This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: Explanatory footnotes are about twice the length of article prose. Move content to main article prose, linked articles, or reduce level of detail. People using screen readers hear explanatory footnotes at the end of the article, meaning they lose context unless there is a small enough number to keep track of mentally. Please help improve this article if you can. (March 2024) (Learn how and when to remove this template message)
← 743 744 745 →
Cardinalseven hundred forty-four
Ordinal744th
(seven hundred forty-fourth)
Factorization23 × 3 × 31
Divisors1, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744
Greek numeralΨΜΔ´
Roman numeralDCCXLIV
Binary10111010002
Ternary10001203
Senary32406
Octal13508
Duodecimal52012
Hexadecimal2E816

744 (seven hundred [and] forty four) is the natural number following 743 and preceding 745.

744 plays a major role within moonshine theory of sporadic groups, in context of the classification of finite simple groups.

Number theory

744 is the nineteenth number of the form where , and represent distinct prime numbers (2, 3, and 31; respectively).[1]

It can be represented as the sum of nonconsecutive factorials ,[2] as the sum of four consecutive primes ,[3] and as the product of sums of divisors of consecutive integers ;[4] respectively:[i]

744 contains sixteen total divisors — fourteen aside from its largest and smallest unitary divisors — all of which collectively generate an integer arithmetic mean of [10][11] that is also the first number of the form [1][ii]

The number partitions of the square of seven (49) into prime parts is 744,[17] as is the number of partitions of 48 into at most four distinct parts.[18][iii]

744 is an abundant number,[23] with an abundance of 432.[24][iv] It is semiperfect, since it is equal to the sum of a subset of its divisors (e.g., 1 + 2 + 4 + 24 + 62 + 93 + 124 + 186 + 248).[33][v]

It is also a practical number,[50] and the first number to be the sum of nine cubes in eight or more ways,[51] as well as the number of six-digit perfect powers in decimal.[52]

Meanwhile, in septenary 744 is palindromic (21127),[53] while in binary it is a pernicious number, as its digit representation (10111010002) contains a prime count (5) of ones.[54][vii]

Totients

744 has two hundred and forty integers that are relatively prime or coprime with and up to itself, equivalently its Euler totient.[5][viii]

This totient of 744 is regular like its sum-of-divisors, where 744 sets the twenty-ninth record for of 1920.[66][ix] Both the totient and sum-of-divisors values of 744 contain the same set of distinct prime factors (2, 3, and 5),[68] while the Carmichael function or reduced totient (which counts the least common multiple of order of elements in a multiplicative group of integers modulo ) at seven hundred forty-four is equal to .[26][x] 744 is also a Zumkeller number whose divisors can be partitioned into two disjoint sets with equal sum (960).[80][xi]

Of these 240 totatives, 110 are strictly composite totatives that nearly match the sequence of composite numbers up to 744 that are congruent to , which is the same congruence that all prime numbers greater than 3 hold.[88][xii] Only seven numbers present in this sequence are not totatives of 744 (less-than); they are 713, 589, 527, 403, 341, 217, and 155; all of which are divisible by the eleventh prime number 31.[xiii][xiv][xv] The remaining 130 totatives are 1 and all the primes between 5 and 743 except for 31 (all prime numbers less than 744 that are not part of its prime factorization) where its largest prime totative of 743[xvi] has a prime index of 132 (the smallest digit-reassembly number in decimal).[127][xvii] On the other hand, only three numbers hold a totient of 744; they are 1119, 1492, and 2238.[5][xviii]

744 is the sixth number whose totient value has a sum-of-divisors equal to : .[131] Otherwise, the aliquot sum of 744, which represents the sum of all divisors of 744 aside from itself, is 1176[14] which is the forty-eighth triangular number,[6] and the binomial coefficient present inside the forty-ninth row of Pascal's triangle.[132][xix] In total, only seven numbers have sums of divisors equal to 744; they are: 240,[xx] 350, 366, 368, 575, 671, and 743.[32][xxi] When only the fourteen proper divisors of 744 are considered, then the sum generated by these is 1175, whose six divisors contain an arithmetic mean of 248,[11][xxii] the third (or fourteenth) largest divisor of 744. Only one number has an aliquot sum that is 744, it is 456.[14][xxiii]

Graph theory

The number of Euler tours (or Eulerian cycles) of the complete, undirected graph on six vertices and fifteen edges is 744.[164] On seven vertices, there are 129,976,320 Euler tours. These can only be generated on complete graphs with at least three vertices; the number of tours for three, four and five vertices are respectively 2, 2 and 264 (the latter is the second digit-reassembly number in base ten).[127] On the other hand, the number of Euler tours of the complete digraph, or directed graph, on four vertices is 256, while on five vertices it is 972,000 (and 247,669,456,896 on six vertices), by the BEST theorem.[165]

Regarding the largest prime totative of 744, there are (aside from the sets that are the union of all such solutions),

Thrice 743 is 2229,[xxiv] whose average of divisors is 744 (as with thrice any prime number , the average of divisors will be ).[10][11][xxv] This value is a difference of 1110 from 3339, which is the sum of seven hundred and forty-two (742)[xxvi] repeating digits of the reciprocal of 743, as the forty-eighth full repetend prime in decimal[25] (with the smallest number to have a Euler totient of 744 being 1119).[5]

For open trails of lengths eight and nine, starting and ending at fixed distinct vertices in the complete undirected graph on five labeled vertices, the number is 132 (the prime index of 743, half 264),[195] that also represents the number of irreducible trees with fifteen vertices.[196][xxvii] While for the complete undirected graph there are 264 directed Eulerian circuits,[201][202] it is more specifically the number of circuits of length ten in the complete undirected graph on five labeled vertices, and as such it is the twenty-fifth element in a triangle of length on labeled vertices.[203]

Otherwise, 745 is the number of disconnected simple labeled graphs covering six vertices, where the most symmetric of these graphs has three pairs of distinct vertices each covered by one edge alone and all three edges intersecting; this yields the disconnected covering graph on vertices labelled through in a hexagonal arrangement, with the remaining 744 graphs representing all other possible conformations.[204]

456 (the only number to have an aliquot sum of 744) is the number of unlabeled non-mating graphs with seven vertices (where a mating or graph is a graph where no two vertices have the same set of neighbors), equivalently the number of unlabeled graphs with seven vertices and at least one endpoint;[205] as well as the number of cliques in the 7-triangular graph, where every subset of two distinct vertices in a clique are adjacent.[206] The number of even graphs with seven vertices, where a graph is odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise, is 456.[207][208][xxviii]

In particular, 456 is the aliquot sum of 264, the only number to have this value for [14][xxix]

Convolution of Fibonacci numbers

744 is the twelfth self-convolution of Fibonacci numbers, which is equivalently the number of elements in all subsets of with no consecutive integers.[225][226][227]

Abstract algebra

j-invariant

The j–invariant holds as a Fourier series q–expansion,

where and the half-period ratio of an elliptic function.[228]

The j–invariant can be computed using Eisenstein series and , such that:

where specifically, these Eisenstein series are equal to and with .

The respective q–expansions of these two Eisenstein series have coefficients whose numerical values are in proportion to 240 and −504, respectively;[229][230] where specifically the sum and difference between the absolute values of these two numbers is 240 + 504 = 744 and 504 − 240 = 264. Furthermore, when considering the only smaller even (here, non-modular) series the sum between the absolute value of its constant multiplicative term (24) and that of (240) is equal to 264 as well. The 16th coefficient in the expansion of is −744, as is its 25th coefficient.[231]

Alternatively, the j–invariant can be computed using a sextic polynomial as: where represents the modular function, with 256 = 28.[232]

Almost integers

Also, the almost integer[233]

This number is known as Ramanujan's constant, which is transcendental.[234] Mark Ronan and other prominent mathematicians have noted that the appearance of in this number is relevant within moonshine theory, where one hundred and sixty-three is the largest of nine Heegner numbers that are square-free positive integers such that the imaginary quadratic field has class number of (equivalently, the ring of integers over the same algebraic number field have unique factorization).[235]: pp.227, 228  has one hundred and ninety-four (194 = 97 × 2) conjugacy classes generated from its character table that collectively produces the same number of elliptic moonshine functions which are not all linearly independent; only one hundred and sixty-three are entirely independent of one another.[236] The linear term of error for Ramanujan's constant is approximately,

where is the value of the minimal faithful complex dimensional representation of the Friendly Giant, the largest sporadic group.[237] contains an infinitely graded faithful dimensional representation equivalent to the coefficients of the series of the q-expansion of the j-invariant. Specifically, all three common prime factors that divide the Euler totient, sum-of-divisors, and reduced totient of are the smallest and only primes that divide the orders of all twenty-six sporadic groups, in contrast with only six groups whose orders can be divided by the largest consecutive supersingular prime and largest prime factor of seven hundred and forty-four, ;[235]: pp.244–246 [α] three of these belong inside the small family of six pariah groups that are not subquotients of [245] The largest supersingular prime that divides the order of is [246][247] which is the eighth self-convolution of Fibonacci numbers, where is the twelfth.[226][β]

The largest three Heegner numbers with also give rise to almost integers of the form which involve . In increasing orders of approximation,[263]: p.20–23 [γ]

Square-free positive integers over the negated imaginary quadratic field with class number of also produce almost integers for values of , where for instance there is [266][267][δ][ε]

D4 and F4

is theta series coefficient of four-dimensional cubic lattice [280][281] On the other hand, in the theta series of the four-dimensional body-centered cubic lattice — whose geometry with defines Hurwitz quaternions of even and odd square norm as realized in the –cell honeycomb that is dual to the –cell honeycomb (and, as a union of two self-dual tesseractic –cell honeycombs) — the sixteenth coefficient is the seven hundred and forty-fourth coefficient in the series;[282] with coefficient index the forty-ninth non-zero norm.[283][ζ]

In four-dimensional space, three-dimensional cell facets of the three-largest of six regular –polytopes (the octaplex, dodecaplex, and tetraplex)[285]: p.292  collectively number [η]

E8 and the Leech lattice

Within finite simple groups of Lie type, exceptional Lie algebra holds a minimal faithful representation in two hundred and forty-eight dimensions, where divides thrice over.[286][287]: p.4  John McKay noted an intersection between finite simple groups of Lie type and those that are sporadic, where symmetries of nodes in the Dynkin diagrams of complex Lie algebra as well as those of and respectively coincide with the three largest conjugacy classes of ; where also the corresponding McKay–Thompson series of sporadic Thompson group holds coefficients representative of its faithful dimensional representation (also minimal at )[288][237] whose values themselves embed irreducible representation of .[289]: p.6  In turn, exceptional Lie algebra is shown to have a graded dimension [290] whose character lends to a direct sum equivalent to,[289]: p.7, 9–11 

where the CFT probabilistic partition function for is of character [291]

The twenty-four dimensional Leech lattice in turn can be constructed using three copies of the associated lattice[292][285]: pp.233–235 [θ] and with the eight-dimensional octonions (see also, Freudenthal magic square),[297] where the automorphism group of is the smallest exceptional Lie algebra , which embeds inside . In the form of a vertex operator algebra, the Leech lattice VOA is the first aside from (as ) with a central charge of , out of a total seventy-one such modular invariant conformal field theories of holomorphic VOAs of weight one.[298] Known as Schellekens' list, these algebras form deep holes in whose corresponding orbifold constructions are isomorphic to the moonshine module ♮ that contains as its automorphism;[299] of these, the second and third largest contain affine structures and that are realized in .[ι][304][305]

Other properties

is also the sum of consecutive pentagonal numbers,[306][307][xxxv]

It is a "selfie number", where [309][310][xxxvi]

is the magic constant of a six by six magic square consisting of thirty-six consecutive prime numbers, between and inclusive.[311][xxxvii]

It is the number of non-congruent polygonal regions in a regular –gon with all diagonals drawn.[319][xxxviii]

There are seven hundred and forty-four ways in-which fourteen squares of different sizes fit edge-to-edge inside a larger rectangle.[321][xxxix]

See also

Notes

Higher arithmetic
  1. ^ 744 is equal to the sum between the 41st and 44th indexed prime numbers, inclusive. Indices 15 and 16 of σ(n) multiply to 240, which is the Euler totient value of 744,[5] and add to 31 which is the largest prime that is not a totative of 744 (less than).
    A sum between 24 and 31 generates the tenth triangular number 55, where the eleventh triangular number is 66.[6] 121 is the sum between these two triangular numbers, which is equivalent to the square of 11. The prime index of 31 and its permutable prime in decimal (13)[7] form the third pair of twin primes (11, 13),[8] whose sum is 24, with respective prime indices 5 and 6[9] that add to 11.
  2. ^ 120 is also equal to the sum of the first fifteen integers, or fifteenth triangular number Σ15
    n = 1     n    ,[6] while it is also the smallest number with sixteen divisors,[12] with 744 the thirty-first such number.[13] This value is also equal to the sum of all the prime numbers less than 31 that are not factors of 744 except for 5, and including 1. Inclusive of 5, this sum is equal to 125 = 53, which is the second number after 32 to have an aliquot sum of 31.[14]
    In the Collatz conjecture, 744 and 120 both require fifteen steps to reach 5, before cycling through {16, 8, 4, 2, 1} in five steps.[15][16] Otherwise, they both require nineteen steps to reach 2, which is the middle node in the {1,4,2,1,4...} elementary trajectory for 1 when cycling back to itself, or twenty steps to reach 1.
  3. ^ The radical 186 = 2 × 3 × 31 of 744[19] has an arithmetic mean of divisors equal to 48,[11] where the sum between the three distinct prime factors of 744 is 62 = 36. 186 is nontotient[20] and noncototient,[21] equal to the number of pentahexes when rotations are counted as distinct.[22]
  4. ^ 744 is the 181st abundant number. Specifically, 432 (twice 216, the cube of 6) has an aliquot sum of 808,[14] where 433 is the thirty-first full repetend prime in decimal that repeats 432 digits, which collectively add to 1944.[25] Less than 1000, the sixtieth and largest such prime number is 983, which repeats 982 digits that sum to 4419, a two-digit dual permutation of the digits of 1944 (and where 2475 = 4419 − 1944 has a reduced totient of 60).[26] 432 is also the twenty-third interprime between twin primes,[27] and the eighth such number whose adjacent prime numbers have prime indices that add to a prime number (167 = 83 + 84, the thirty-ninth prime);[9] 348, the composite index of 432,[28] is itself the sixth interprime whose adjacent prime numbers have indices that add to a prime number (139 = 69 + 70, the thirty-fourth prime).[9]
    Importantly, 432 = 33 + 43 + 53 + 63[29] is an Achilles number, a powerful number (the thirty-fourth)[30] that is itself not a perfect power, like 864 (the eleventh index, that is twice its value), and 288 (the totient of 864),[5] as well as 108 (a fourth of 432), and 1944.[31] (432 is the sixth member, where the sixth triangular number is 21,[6] itself the index of 1944 in the list of Achilles numbers. Notice also that 1944 = 1200 + 744, where 1200 = 456 + 744, or in other words σ(456).)[32]
    On the other hand, 181, the index of 744 as an abundant number, is the sixteenth full repetend prime in base ten,[25] and the composite index of 232,[28] that is the sum of the first five interprimes to lie in-between twin primes with prime sums from respective indices (i.e., 138, 72, 12, 6, 4), as with 432 and 348.[27] 109 is the tenth full repetend prime, repeating 108 digits whose sum of repeating digits is 486, a fourth of 1944, where 487 is the thirty-third full repetend prime (and 811 the forty-ninth, one less than the sum of the 180 repeating digits of 181; with 288 = 180 + 108).[25]
  5. ^ 744 is the 183rd semiperfect number, an index value whose sum-of-divisors is 248,[32] and an arithmetic mean of divisors equal to 62 = 31 × 2,[10][11] both of which are divisors of 744 (fourteenth and tenth largest, respectively); otherwise, the sum of its two divisors greater than 1 is 3 + 61 = 64 = 82. 183 is also the eighth perfect totient number,[34] and the number of semiorders on four labeled elements.[35] It is the largest number of interior regions formed by fourteen intersecting circles, which is equivalent to the number of points in the projective plane over the finite field , since 132 + 13 + 1 = 183.[36] It is the number of toothpicks in the toothpick sequence after eighteen stages.[37] Following, 181, 183 is the sixty-second Löschian number of the form x2 + xy + y2, as a product of the third and twenty-fourth such numbers (3, 61), of prime indices two and eighteen.[38][9] Furthermore, the smallest number to have exactly four solutions to this quadratic polynomial is the first taxicab number 1729, from (a,b) integer pairs (3, 40), (8, 37), (15, 32) and (23, 25),[39][40] where these four pairs collectively generate a sum of 183. The smallest such number with only two solutions, on the other hand, is 49;[41] wherein 1729 is the 97th such number expressible in two, or more, ways.[42] The probability that any odd number is a Löschian number is 0.75, while the probability that it is an even number is a remaining 0.25.[43]
    There are precisely 816 integers (uninclusively, equal to the sum-of-divisors of 737, the largest composite totative of 744)[32] between the semiperfect index of 744 and 1000, a number that in-turn has a sum-of-divisors of 2232 = 744 × 3,[32] the 24th decagonal number,[44] as well as the 30th number that can be expressed as the difference of the squares of consecutive primes in just one distinct way;[45] other members in this sequence include:
    • 1920 (26th), the sum-of-divisors of 744,[32]
    • 1728 (24th), equal to the cube of 12, and one less than the first taxicab number,
    • 432 (8th), the abundance of 744,[24]
    • 288 (6th), the difference between 744 and 456, the only number to have an aliquot sum of 744,[14]
    • 240 (5th), the Euler totient of 744.[5]
    888 is the 13th member, the 26th repdigit in decimal[46] equal to the sum between 432 and 456, and between 144 and 744 (where 432 and 456 both have totient values of 144);[5] and equal to the product between the twelfth prime number 37, and 24. The first four members (5, 16, 24, 48) generate a sum of 93, which is the eleventh largest divisor of 744, following 62.[45] 93 represents the number of different cyclic Gilbreath permutations on 11 elements,[47] and consequently there are ninety-three different real periodic points of order 11 on the Mandelbrot set.[48] Otherwise, 1632 = 1176 + 456, in equivalence with the aliquot sum of 744 (1176)[14] and 456, is the sixteenth number that can be expressed as the difference of the squares of primes in just two distinct ways, consecutive or otherwise (432, 1728, 1920 and 2232 are also in this sequence).[49]
    183 is also the first non-trivial 62-gonal number.
  6. ^ Meanwhile, where 279 is the 58th interprime to lie between consecutive odd numbers that are not necessarily twin primes (here at a difference of four), the 58th palindromic number in septenary is the equivalent of 456 in decimal[53] (that is the only number to have an aliquot sum of 744)[14] and represented as 12217, whose digits are the dual permutation of the digits of 744 as the 64th palindrome in base-7 (2112).[53]
    Explicitly, every positive integer is at most the sum of two hundred and seventy-nine eighth powers (Waring's problem), preceded by a maximum number of {1, 4, 9, 19, 37, 73, 143} n-powers;[60] where, as aforementioned, 279 is the decimal representation of the ones' complement of 744 in binary. In this sequence, the first six members generate a sum equivalent to the seventh member 143, the composite index of 186[28] that is the radical of 744,[19] as well as equal to the sum of seven consecutive primes starting from 11 through the eleventh prime number: 11 + 13 + 17 + 19 + 23 + 29 + 31. It is also the product between the third twin prime pair (11 × 13);[8] while figuring in 34 + 44 + 54 + 64 = 74 − 143, which is the second (or third) exception in the sequence of polynomials that starts with 32 + 42 = 52 and 33 + 43 + 53 = 63, after 31 = 41 − 1 (and, 0 = 30 − 1).
    While in the beginning of the sequence of full repetend primes (with primitive root 10) the two smallest such numbers are 7 and 17, with the repeating digits of the reciprocal of 7 adding to 27 = 33 that is the seventeenth composite number,[28] there is also 983 + 17 = 1000, and 2 × 279 + 432 = 7 + 983. The span of full repetend primes between 17 and 983 is of fifty-nine integers, where 59 is the seventeenth prime number, and seventh super-prime.[9][61]
  7. ^ 744 is the four hundred and sixth indexed pernicious number, where 406 is the twenty-eighth triangular number;[6] in its base-two representation, the digit positions of zeroes are in 1:1:3 or 3:1:1 ratio with the positions of ones, which are in 1:3:1 ratio.
    Its ones' complement is 1000101112, equivalent to 279 = 32 × 31 in decimal, which represents the sum of GCDs of parts in all partitions of 16 = 42.[55] It is also the number of partitions of 62 = 2 × 31 (a divisor of 744) as well as 63 into factorial parts (without including 0!),[56] and the number of integer partitions of 44 whose length is equal to the LCM of all parts[57] (with 63 the forty-fourth composite number,[28] where 44 is itself the number of derangements of 5,[58] and 63 + 44 = 107 the twenty-eighth prime number).[9]
    211210 is the number of repeating decimal digits of 2113 as a full repetend prime in decimal,[25] where 2112 is the 65th interprime to lie between consecutive twin primes,[27] otherwise it is the 317th member between consecutive odd primes,[59] where 317 is the 66th indexed prime number (with 90 and 91 the 65th and 66th composites, respectively, where 181 = 90 + 91).[9][28] 279, on the other hand, is the 58th number to lie between consecutive odd prime numbers (277, 281),[59] with 81 = 92 the fifty-eighth composite number.[28][vi] More specifically, eighty-one is the sum of the repeating digits of the third full repetend prime in decimal, 19,[25] where the sum of these digits is the magic constant of an 18 × 18 non-normal yet full prime reciprocal magic square based on its reciprocal (1/19).[62][63][64]
  8. ^ 744 is the twenty-third of thirty-one such numbers to have a totient of 240, after 738, and preceding 770. The smallest is 241, the fifty-third prime number and sixteenth super-prime,[61] and the largest is 1050, which represents the number of parts in all partitions of 29 into distinct parts.[65]
  9. ^ The value of this sigma function represents the fifteenth sum of non-triangular numbers in-between triangular numbers; in this instance it is the sum that lies in-between the fifteenth (120) and sixteenth (136) triangular numbers[67] (i.e. the sum of 121 + 122 + ... + 135).
  10. ^ The thirtieth triangular number is 465 = 3 × 5 × 31, equal to the difference between 744 and 279 (equivalently 10111010002 and 1000101112, ones' complement pairs respectively). 30 is the twelfth number m such that 6m + 1 and 6m − 1 are twin primes (181, 179).[8][69] 465 is also the binomial (31,2) equal to the number of size-2 subsets of {0, 1, ..., 31} that contain no consecutive integers (in light as a triangular number, trailing its sequence by one index).[70][6] In the Padovan sequence, 465 is the twenty-eighth indexed member equal to the number of compositions of 28 into parts congruent to 2 mod 3, also the number of compositions of 28 into parts that are odd and greater than or equal to 3, and the number of maximal cliques in a (28 + 6)–path complement graph (specifically, in the 34–path).[71]
    30 is the smallest sphenic number that is the product of three distinct prime numbers (2, 3, and 5), with the next two such numbers 42 + 66 = 108.[72] Importantly, 30 is the unique point in the sequence of natural numbers where the ratio of prime numbers to non-primes (up to) is 1/2; when including 0 in + ∪ {0} = 0 as a non-prime, this ratio is reached at 29, where a ratio of one-half is only reached again (here, purely as a ratio between primes and composites) at 37 and 43, which are respectively the twelfth and fourteenth primes (when including 1 as a prime unit, then the thirteenth prime number 41 would be another point of 1:1 ratio, without taking 0 into account).[9] Otherwise, a 1:1 ratio of primes to composites (up to) occurs at two points: 11 and 13, respectively the fifth and sixth prime[9] — whose indices sum to 11, the prime index of 31 — and at 7, when including 0 and 1 as non-primes in , or at 8 when only including 1, and at 9 when considering 0 as non-prime and 1 as a prime unit). Thirty is also the first composite value with three distinct prime factors or more for which the Möbius function returns −1, as with all numbers with a strictly odd number of prime factors, including prime numbers themselves. Relatedly, 31 is the first number (after 1) to reach a record amplitude between zeros in the associated Mertens function M(n), where the next such number is 114,[73] whose arithmetic mean of divisors is 30,[10][11] with a sum-of-divisors of 240.[32] On the other hand, 456 (the only number to have an aliquot sum of 744),[14] 744 and 1176 (the aliquot sum of 744) all set a value of −5 for M(n),[74] respectively the 20th, 57th, and 86th numbers to do so (where 57 is the fortieth composite, and 80 the fifty-seventh),[28] with these indices generating the sum 20 + 57 + 86 = 163, in equivalence with the largest of nine Heegner numbers. (Also, 20 + 57 = 77, the sum of base ten digits in the almost integer approximation of eπ√163 before its decimal expansion.)
    The sixth pair of numbers to generate a record prime gap (of 14) are the 30th and 31st prime numbers (113, 127),[75][76][77][9] The previous consecutive primes to generate record gaps are (89, 97), (23, 29), (7, 11), (5, 3) and (3, 2);[78][79] the lesser of these together generate a sum of 127, the thirty-first prime number.
  11. ^ It is the 168th indexed Zumkeller number, where 168 = 6 × 28 represents the product of the first two perfect numbers,[81] equal to the fifth Dedekind number,[82] where the previous four indexed members (2, 3, 6, 20) collectively add to 31 — with 496 the thirty-first triangular number,[6] and third perfect number. 168 is also the number of primes below 1000.[9][83] The two sets of divisors of 744 with equal sums are:
    • (1, 2, 3, 4, 8, 12, 62, 124, 744)
    • (6, 24, 31, 186, 93, 248, 372)
    960 is also the thirty-first Jordan–Pólya number that is the product of factorials 5! × (2!)3,[84] equal to the sum of six consecutive prime numbers 149 + 151 + 157 + 163 + 167 + 173, between the 35th and 40th primes (it is the thirty-fifth such number).[85] The fifteenth and sixteenth triangular numbers generate the sum 120 + 136 = 256 = 28 that is the totient value of 960,[5] and the number of partitions of 29 = 1 + 2 + 3 + 5 + 7 + 11 = 22 + 32 + 42 into distinct parts and odd parts.[86] Like its twin prime 31, 29 is a primorial prime, which together comprise the third and largest of three known pairs of twin primes (1st, 2nd, and 5th) to be primorial primes,[87] with possibly no larger pairs. In between these lies 30.
  12. ^ The composite totatives are
    {25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 161, 169, 175, 185, 187, 203, 205, 209, 215, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 343, 355, 361, 365, 371, 377, 385, 391, 395, 407, 413, 415, 425, 427, 437, 445, 451, 455, 469, 473, 475, 481, 485, 493, 497, 505, 511, 515, 517, 529, 533, 535, 539, 545, 551, 553, 559, 565, 575, 581, 583, 595, 605, 611, 623, 625, 629, 635, 637, 649, 655, 665, 667, 671, 679, 685, 689, 695, 697, 703, 707, 715, 721, 725, 731, 737}.
    Its smallest composite totative is 52 = 25, that is the only number to have an aliquot sum equal to 6,[14] and the second of only two numbers to have a sum-of-divisors of 31 (the largest prime factor of 744), after 16 = 42.[32] Specifically, the twenty-fifth prime number 97 is generated from a sum of 48 and 49. Ninety-seven is the ninth full repetend prime in decimal,[25] whose 96 repeating digits generate a sum of 432, equal to the abundance of 744.[24] 97 is also the first odd prime number that is not a cluster prime[89] (which precedes the thirty-first prime number 127 in this sequence), where the prime gap for a cluster prime is six or less;[90] the sixth and seventh primes that are not cluster primes are the forty-eighth and forty-ninth primes 223 and 227.
    97 is the eleventh prime number p of the form 6m + 1 for a natural number m.[91]
    Between the median composite totative values of 744, i.e. (413, 415), is 414 — the 333rd composite number[28] — with an Euler totient of 132[5] that is also the prime index of 743,[9] the largest prime totative of 744. Furthermore, 414 has a sum-of-divisors of 936,[32] the 777th composite number, in-turn the sum-of-divisors of 639 (palindromic with 936), itself the composite index of 777 (also palindromic). Where 414 is the 43rd number to return 0 for the Mertens function, 333 is the 26th,[92] with 43 the fourteenth prime number.[9] Also, the twenty-fifth repdigit in decimal is 777.[46]
    Where 25 is the smallest composite totative of 744, the largest such totative is 737, whose aliquot sum is 79,[14] the twenty-second indexed prime number,[9] that is the lesser permutable prime to 97 in decimal.[7]
  13. ^ 713 is the product of the ninth and eleventh prime numbers 23 × 31, where the twenty-third prime number (83) is the eleventh prime of the form 6m − 1; it is equal to the difference between 744 and 31, with an aliquot sum of 55[14] and totient of 660.[5]
    ∗ 589 equal to 19 × 31 is the sum of three consecutive primes (193 + 197 + 199).[93] It is also the ninth centered tetrahedral number, where 5 and 15 are the first two such numbers (121 = 112 is the fifth).[94] It is the fourteenth third spoke of a hexagonal spiral,[95] and the twentieth quasi-Carmichael number,[96] with a reduced totient of 90.[26] In the spt function, 589 is the total number of smallest parts (counted with multiplicity) in all partitions of 15.[97]
    ∗ 527 equal to 17 × 31 is the number of partitions of 31 with equal number of even and odd parts.[98] Its aliquot sum is 49[14] whose arithmetic mean of divisors is 122 = 144, with a sum of its prime factors equal to 48. It is also the maximal number of pieces that can be obtained by cutting an annulus with 31 cuts,[99] and equivalent to the sum of thirty-one consecutive non-zero integers 2 + 3 + ... + 32; the second-smallest such sum after the third perfect number 496.[81]
  14. ^ ∗ 403 equal to 13 × 31 is the thirty-seventh number to return 0 for the Mertens function,[92] and the value of the sum-of-divisors of 144, the first number to generate this value.[32] It is also the first number to hold a totient of 360, of twenty-five such numbers; with 549 the sixth (and the 447th composite number, a value that is itself the 360th composite)[28] and 693 = 144 + 549 the tenth.[5] Also, 403 is the 284th arithmetic number, since its four divisors hold an arithmetic mean of 112 = 7 × 4 × 4, the second such number to hold this value.[10][11]
    The 112nd prime number is 613,[9] which is the sum of successive composite indexes of 382 and 384 (respectively, 306 and 307),[28] where 383 is the 28th full repetend prime number in decimal; 613 is also equal to the sum of prime indices of seven consecutive full repretend prime numbers (in decimal): 389, 419, 433, 461, 487, 491, 499, respectively with prime indices 77, 81, 84, 89, 93, 94, and 95[9] (and between the 29th and 35th such primes)[25] — all following 383, where the next full repetend prime after 499 is the 96th prime number 503.[9] 383 is also the second number, after 19, to generate a full prime reciprocal magic square in decimal whose rows and diagonals generate the same magic constant (from its repeating digits) of 1719.[62] Within the list of prime numbers, 2017 and 2027 are respectively the 306th and 307th,[9] where the 1719th composite number is 2026.[28]
    • The arithmetic mean of divisors of 2022, which lies midway between the prime gap of eleven integers (inclusive) just noted, is 507, equal to the sum of prime factors of 2012 = 22 × 503 (inclusive of multiplicities), whose arithmetic mean of divisors in-turn is 588 (half 1176, the aliquot sum of 744) and the second such number to have this value (588), after 1304.[10][11]
    • 1685, on the other hand, is the third number to have an average of divisors that is 507 after 2022, and whose sum-of-divisors (of 1685) is 2028,[32] a number lying between the 307th and 308th prime numbers, 2027 and 2029, the 62nd pair of twin primes);[9][8] 2011 is prime as well, the 305th prime number, with 305 having a composite index 242.[9][28]
    • 1715, the composite index of 2022,[28] is the fourth of thirteen numbers to hold a totient of 1176 = s(744),[5] with an arithmetic mean of divisors equal to 300, which is the twenty-fourth triangular number.[6]
    • 2023, midway between primes 2017 and 2029, has an aliquot sum of 433 (or 432, the abundance of 744, when not counting 1),[14] and a Euler totient of 1632 = 1176 + 456 (the aliquot sum of 744, plus the only number to hold an aliquot sum of 744).[5]
    Where 383 repeats 382 digits in base ten, 382 is an arithmetic number too, specifically the first number to have an average of divisors of 144[10][11] while being the seventh number to hold a sum-of-divisors of 576 = 242.[32] Also, 613 + 744 = 1357, alongside 744 − 613 = 131 (both with successive differences of 2 between digits); the latter is the twelfth full repetend prime number in decimal that repeats 130 digits with a sum of 585 = 80 + 81 + 82 + 83, where the 48th full repetend prime, on the other hand, is 743, the largest prime totative of 744 (or 49th in the near-identical sequence of long primes, that instead begins with 2, whose decimal expansion of 1/p has period p − 1).[25][100]
    Meanwhile, 112 (the 82nd composite) is the composite index of 147 = 48 + 49 + 50[28] with an aliquot sum of 81 = 92[14] and average of divisors of 38.[10][11] 147 itself is the composite index of the nineteenth triangular number, 190,[6] equal to the totient and reduced totient of 382.[5][26] 744 − 383 = 361 = 192, where 383 is the 76 = 22 × 19 indexed prime number[9] (and 304 fourfold seventy-six, the prime index of 2003, which is the smallest prime number greater than 2000).[9]
    ∗ 341 equal to 11 × 31 is the sum between seven consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61), of prime indexes 12 through 18. It is the smallest Fermat pseudoprime to binary,[101] and the sixth centered cube number.[102] It is the number of partitions of 31 such that every part occurs with the same multiplicity,[103] and the thirty-first number in the Moser–de Bruijn sequence to be a sum of distinct powers of 4 (44 + 43 + 42 + 41 + 40).[104][105] 341 is also the eleventh octagonal number,[106] equal to the number of nodes in a regular eleven-sided undecagon with all diagonals drawn.[107] s(341) = 43,[14] with an arithmetic mean of divisors equal to 96.[11] It is also the tenth Jacobsthal number,[108] and the total number of largest parts (or equivalently sum of smallest parts) in all partitions of 16.[109]
  15. ^ ∗ 217 equal to 7 × 31 is the ninth centered hexagonal number, which also includes as previous members {7, 19, 37, 61, 91, 127, 169}[110] where 127 is the sixth member that is also the thirty-first prime number. 217 is also the sixth non-trivial dodecagonal number[111] and third non-trivial centered 36-gonal number.[112] It is the third Fermat pseudoprime to quinary after 124 (a divisor of 744) and 4,[113] and the fifteenth Blum integer whose distinct prime factors are congruent to 3 mod 4.[114]
    ✶ 155 equal to 5 × 31 has an aliquot sum of 37[14] and an arithmetic mean of divisors equal to 48.[5] It is the third number equal to the sum from its lowest prime factor through its largest, the only smaller numbers with the same property are 10 and 39[115] which generate a sum of 49 = 72; the next such number is 371, 1 less than half of 744. There are one hundred and fifty-five total primitive permutation groups (G, X) of degree 81 that only preserve partitions on the set X by G that are trivial;[116] the sum of the number of all permutation groups of lower degrees is 666, which is doubly triangular,[117] since it represents the sum of the first thirty-six integers, where 36 is itself the eighth triangular number.[6] The 148 repeating digits of the thirteenth full repetend prime number in decimal 149 (that lies between 109 and 181 in this sequence) [25] sum to 666.
    The one hundred and fifty-fifth indexed practical number is 744,[50] where the 155th arithmetic number is the forty-eighth prime number 223: the first number to hold an arithmetic mean of divisors of 112, preceding 403.[10][11]
  16. ^ 743 is the twenty-sixth or twenty-seventh lucky number of Euler of the form n2 + n + 41 to yield consecutive prime numbers.[118][119] It is also the twenty-sixth indexed beginning of a maximal chain of primes of the form p(k), ..., p(k+r) for r ≥ 1.[120] It is the forty-eighth full repetend prime in decimal with primitive root 10,[25] or similarly the forty-ninth long-period prime when including 2 in this sequence.[100] It is the forty-eighth member in base ten such that every suffix is prime, in which repeatedly removing the most significant digit yields a prime number at every step, until a single-digit prime number remains.[121]
    Specifically, 743 is the total sum of odd parts in all partitions of 13,[122] and the number of partitions of 31 into partition numbers.[123] Particularly, it is the number of ways to partition 7 labeled elements into sets of sizes of at least 2, and order the sets,[124] and furthermore, the number of strict integer partitions of 37 with no part divisible by any and all the other parts.[125]
    Where 743 is the largest prime totative of 744, it represents the sum of all parts of all partitions of all positive integers less than or equal to 25 into an odd number of equal parts or, equivalently, consecutive parts;[126] with twenty-five the smallest composite totative of 744.
  17. ^ These prime totatives generate a sum of 44,647 (a prime count, the 4641th prime number, an index whose arithmetic mean of divisors is 504)[11] while the composite totatives collectively generate a sum of 44,632 (one less than a prime number, the 4639th prime and 626th super-prime,[61] the latter an index whose value is the fourth and largest number to have an aliquot sum of 316);[14] these sums have a difference of 15, or otherwise collectively have a range of 16 numbers between them, inclusive.
    On the other hand, differences and sums between the six numbers congruent ±1 mod 6 that are not totatives of 744 are themselves equivalent to divisors of 744, since they are biprimes in proportion with 31 (for example 713 − 589 = 124 and 713 − 527 = 186), while also generating other relevant equalities such as 713 − 217 = 496.
    The greatest difference between the largest (713) and smallest (155) of these totatives is 558, the seventh number such that the sum of largest prime factors of numbers from 1 to 558 is divisible by 558, the previous and sixth indexed number is 62,[128] the tenth-largest divisor of 744, whose sum-of-divisors is equal to 96;[32] the fifth number in this sequence is 32 (which divides 96).
    The difference between 558 and 62 is also 496, equal to 713 − 155 − 62; and importantly, 558 is twice 279, which is the ones' complement of 744 in binary in its decimal representation.
  18. ^ The difference between 2238 and 1492 is 746, where 2238 is twice 1119. Specifically, 746 = 2 × 373 = 15 + 24 + 36 = 2! + 4! + 6! is nontotient,[20] and equal to the number of non-normal orthomagic squares with magic constant of 6.[129]
    Its totient value of 372 is half of 744,[5] that is also 1 less than 373, the 74th indexed prime number[9] that is the smallest difference between numbers that have totients equal to 744 (i.e., 1119 − 1492). 373 is the eleventh two-sided left-and-right truncatable prime number in decimal, of a total fifteen such primes in base ten.[130] While 373 is the smallest difference between numbers that have a totient value of 744, its digit representation in decimal is the mirror permutation of the digits of 737, the largest composite totative of 744.
    The sum of these three numbers whose totients equal 744 is 1119 + 1492 + 2238 = 4849.
  19. ^ 1176 is also one of two middle terms in the twelfth row of a Pascal triangle.[133] In the triangle of Narayana numbers, 1176 appears as the fortieth and forty-second terms in the eighth row,[134] which also includes 336 (the totient of 1176)[5] and 36 (the square of 6). Inside the triangle of Lah numbers of the form , 1176 is a member with n = 8 and k = 6.[135] It is a self-Fibonacci number; the fifty-first indexed member where in its case divides ,[136] and the forty-first 6-almost prime that is divisible by exactly six primes with multiplicity.[137]
  20. ^ 240 is the thirty-first quarter square, where 49 is the 14th.[138] Relatedly:[138]
    • The maximum product of two integers whose sum is 31 is 240.
    • 240 is the sum of the positive integers less than 31 that have the opposite parity as 31.
    • 240 is the number of nonempty subsets of {1, 2, ..., 31} that contain exactly one odd and one even number.
    • There are 240 arithmetic progressions of three terms by any means which can be extracted from the set of the first thirty-one natural numbers (starting from 1).
    From the starting position in standard chess, 240 is the minimum number of captures by pawns of the same color to place 31 of them on the same file (column). 240 is also the maximal number of edges that a triangle-free graph of 31 vertices can have.[138]
    Importantly, 240 is the polynomial degree of a sixteen-cycle logistic map r16,[139][140][141] where 16 is the smallest number to have an aliquot sum of 31.[14]
  21. ^ The sum of all these seven integers whose sum-of-divisors are in equivalence with 744 is 3313, the 466th prime number[9] and 31st balanced prime,[142] as the middle member of the 49th triplet of sexy primes (3307, 3313, 3319);[143] it is respectively the 178th and 179th such prime p where p − 6[144] and p + 6[145] are prime — where 178 is the 132nd composite number,[28] itself the prime index of the largest number (743) to hold a sum-of-divisors of 744.[9][32] 3313 is also the 24th centered dodecagonal or star number[146] (and 15th that is prime).[147] Divided into two numbers, 3313 is the sum of 1656 and 1657, the latter being the 260th prime number, an index value in-turn that is the first of five numbers to have a sum-of-divisors of 588,[32] which is half of 1176 (the aliquot sum of 744);[14] its arithmetic mean of divisors (of 260), on the other hand, is equal to 49.[10][11] Furthermore, 260 is the average of divisors of 2232 (the fourth largest of five numbers to hold this value), which is thrice 744.
  22. ^ 248 is twice 124 and half 496; the latter which is the third perfect number[81] — like 6 and 28 (twice 14) — that is also the thirty-first triangular number,[6] with form 2p - 1(2p − 1) and p = 5 by the Euclid-Euler theorem. The totient of 248 is 120,[5] with eight divisors that produce an integer arithmetic mean of 60;[11] 496 has a totient of 240 like 744.
    Where the sum between 744 and the totient 256 = 28 of 960 (the Zumkeller half of 744) is equal to 1000 = 103, their difference is equivalent with 488 = 240 + 248.
    248 is also the sum of prime indices of seven consecutive full repetend prime numbers in decimal: 109, 113, 131, 149, 167, 173, 181,[25] respectively with 29, 30, 32, 35, 39, 40, and 42 prime indices;[9] where 248 is the one hundred and ninety-fourth composite number, and 194 the 149th composite number,[28] the middle term in this consecutive sequence of seven full repetend primes. 149 itself is a strong concatenation of 14 = 7 + 7 and 49 = 7 × 7, joined in the middle by 4, where 7 is the fourth prime number. In the list of successive prime and composite indices of 181, starting with its prime index of 42, the sum generated is 109 = 42 + 28 + 18 + 10 + 5 + 3 + 2 + 1 + 0; with 109 the composite index of 144.[9][28]
  23. ^ 456 is the sixth icosahedral number,[148] one less than the 88th prime number 457 (itself an index that represents the 64th composite).[9][28] The list of icosahedral numbers includes 124 as the fourth indexed member, where this number is also the third non-unitary stella octangula number after 51 and 14,[149] and the seventh untouchable number, after 120 and 96;[150] it is the eleventh non-unitary divisor of 744, equal to the sum of all the prime numbers that are not part of its prime factorization, less than 31 (i.e. 5 + 7 + ... + 29). It is furthermore the fourth perfectly partitioned number after 1, 2 and 3.[151] Where 1176 and 456 collectively have a range of seven hundred and twenty-one integers, the difference between 721 and 1000 is 279 (the decimal equivalent of the ones' complement of 744 in binary); where also, 1455 = 1176 + 279 holds an arithmetic mean of divisors that is 294,[10][11] which is one-fourth 1176. This sum also represents the number of permutation groups of degree 6,[152] and the number of partitions of 48 into distinct parts such that the number of parts is odd,[153] and the number of partitions of 48 into distinct parts such that the number of parts is even.[154] 456 is also, specifically, the number of partitions of the twenty-ninth composite number 44 into prime parts (where on the other hand, 744 the number of partitions of 49 into prime parts).[17]
    Furthermore, regarding twenty-nine:
    • the sum of the first ten prime numbers up to 29 yields 129,[155] which is the number of prime totatives of 744 (with the fifth such sum of ten consecutive prime numbers being 264, or twice 132). 129 is also the 97th indexed composite number,[28] where 97 is the 25th prime number,[9]
    • the twenty-ninth prime number, on the other hand, is 109,[9] which is the number of composite totatives of 744 greater than 25 = 52, its smallest composite totative;
    • the twenty-ninth record for the sum-of-divisors of a number n is set by σ(744) = 1920.[66]
    The count of prime and composite totatives of 744, sans including 1, is 239, which is the 52nd indexed prime number.[9] It is also the sum of prime pairs (23, 29) and (89, 97) that respectively generate consecutive record prime gaps of 6 and 8, which precede the record gap of 14 (the sum of the previous two, which also multiply to 48) between the thirtieth and thirty-first prime numbers (113, 127),[75][76][77][78][79] which sum to 240, the Euler totient of 744.[5]
    Of highlight, 239 is the only number in concomitance to require a maximum number of squares (four), a maximum number of cubes (nine), and a maximum number of fourth powers (nineteen) to express it[156] — the only other number to require nine cubes is the ninth prime number 23, which is the index of 744 in the list of thirty-one integers to hold a Euler totient value of 240;[5] 23 and 239 are also the third and fourth numbers n (after 1, 2, and 5) that represent the number of partitions of 3n, here 33 and 34 respectively, into powers of 3.[157] When 239 is multiplied with the 628th prime number (4649),[9] it yields the seventh repunit in decimal (1111111),[156] a number whose arithmetic mean of divisors is 279000 = 279 × 1000.[10][11] 628 is the 52nd untouchable number, where 52 is the third, following 2 and 5.[150]
    While the only two positive square stella octangula numbers are 1 and 9653449 = 31072 = (13 × 239)2,[158] — of indexes 1=12 and 169 = 132 in this sequence[149] — the only solutions to y2 + 1 = 2x4 in positive integers are (x, y) = (1, 1) and (13, 239).[159] The fraction 239/(132) is the seventh approximation of the converging continued fraction for √2,[160] where 16 arctan (1/5) − 4 arctan (1/239) is an approximation for π radians.[156]
    5 × 239 = 1195, the fourth number to have an arithmetic mean of divisors equal to 360,[10][11] where specifically, 52 = 13 × 4 = 26 × 2, the prime index of two hundred and thirty-nine, is the fifth Bell number that counts the number of ways to partition a set of five labeled elements.[161] Three-hundred and sixty, the smallest number with twenty-four divisors[12] and fifty-second friendly number (the largest known "small" friendly number is 372, or half 744),[162] is the sixth of only seven (highly composite) numbers such that no number less than twice as much has more divisors.[163] It is also divisible by all integers less than or equal to ten, except for seven, itself a sum of integers that yields 48.
  24. ^ In the family of directed multigraphs that are enriched by the species of set partitions, the number of such multigraphs with four labeled directed edges (or arcs) is 2229.[171][172] On the other hand,
    • 2229 is equal to the sum between the sixteenth Lucas number and its counterpart non-Lucas number, plus 1: (2207, 22).[173][174][175]
    • 2229 is also the number of partitions of 31 with at most 3 odd parts,[176] and the number of partitions of the sixteenth composite number 26[28] that contain a distinct numbers of odd and even parts.[177]
    • It is also the eighth member n of polyhexes of size 6n + 1 = 49 that have 6-fold rotational symmetry, preceding such polyhexes of sizes (1, 1, 2, 4, 11, 37, 136, 540) for 0 ≤ n < 8;[178] for the number of polyhexes of size 31 (n = 6) there are thirty-seven, while for those of size 37 (n = 5) there are 136, which is the sixteenth triangular number.[6]
    2229 is also the number of endofunctions of class with distinct cycle lengths.[179]
    When grouping successive composite numbers together such that the sum of the –th group is a multiple of the –th prime, 2229 is in equivalence with the twenty-fifth group-sum of composites as divided by twenty-fifth prime number, 97.[180] This sum occurs between 182 composite numbers that span the range (1106, 1313), without including the twenty-six prime numbers in between (starting with 1109 and ending with 1307).
    The first composite member of the composite grouping is 1106, the fourteenth 14-gonal number,[181][182] that is also the smallest k such that |M(k)| = 14 (a.k.a. the inverse of the Mertens function).[183] Where 1106 is the number of regions into which the plane is divided when drawing twenty-four ellipses,[184] the number of inequivalent Hadamard matrix designs of order 24 is 1106; explicitly, of form 2–(23, 11, 5).[185][186] Also, the first prime number within the sequence of twenty-five consecutive composite numbers that are divisible by the twenty-fifth prime number is 1109, the 186th prime number (an index that represents the thirteenth largest divisor of 744),[9] that is the seventh number such that the Mertens function reaches a new record amplitudes between zeros (of negative 15);[187] the largest prime number within this sequence is the 214th prime, 1307.[9] 1106 also has a sum-of-divisors of 1920,[32] and the arithmetic mean of its divisors is 240,[10][11] respectively the sum-of-divisors and totient of 744.[5]
    In the list of successive composite indexes — A(n + 1) = A(n)–th composite with A(1) = 47 (cf. A059408) — 1106 is part of the sequence (47, 66, 91, 122, 160, 207, 264, 332, 413, ..., 1106, ...); a sequence rooted in the fifteenth prime number, 47.[9] The square of 13 is 169, and the square of 14 is 196; the latter is 1 and 3 units away from 197 and 199, which are the smallest k such that |M(k)| = n, for n of 7 and 8 (respectively),[183] where the sum between the fifteenth pair of twin primes[8] 197 + 199 = 396 is the third digit-reassembly number in decimal[127] equal to the sum of the first two (132, 264).
  25. ^ Another relevant example is the thirtieth prime number 113,[9] where thrice its value is 339, whose average of divisors is 114.[10][11] The three largest of a total four numbers that have a sum-of-divisors of 456 (the only number to have an aliquot sum of 744)[14] have arithmetic means that are 114 (they are 222, 302, 339, and 407),[10][11] with 339 between adjacent terms that sum to the 127th prime number and 31st super-prime, 709.[61] In the list of numbers to reach record amplitude values between zeroes in the Mertens function, the first few terms are (1, 31, 114, 199, 443, 665, 1109, ...);[187] the fourth term is the 46th prime number (199),[9] an index that is itself the 31st in the list of composites,[28] with 443 having an arithmetic mean of divisors of 222 (since it is prime).[10][11] After 1109, the next term is 1637, which is 66516 in hexadecimal; 31 is the sum-of-divisors of 16, the first of only two numbers (the other being 25),[32] where 31 is also the sum of prime factors of 665 = 5 × 7 × 19.[188] Otherwise, the difference 709 − 339 = 370 represents the second-largest non trivial Armstrong number equal to the sum of the cubes of their digits (of only four in decimal, aside from 0 and 1),[189][190] which follows 153, where 407 is the largest; else, 709 and 339 have a range of 371 integers, which is the third-largest Armstrong number equal to the cube of its digits.
    The 153rd non-trivial and unitary arithmetic number after one is 222, where (153, 154) form the sixth Ruth-Aaron pair[191] with a common sum of distinct prime factors of 20[192] (the eleventh composite, where the first Ruth-Aaron pair is between 5 and 6). The sum of the composite index of 20, alongside the twentieth composite and prime numbers, is in equivalence with 11 + 32 + 71 = 114. Where thrice 20 is 60, half 114 is 57, which represents the composite index of 80, with 57 the 40th composite number.[28] Ten is the sum between sums of distinct prime divisors (of 5) of the first two Ruth-Aaron pairs, where the second pair is (24, 25), followed by (49, 50).[191]
  26. ^ 742 is the seventh icosahedral number, following 456.[148] The sum of all non-unitary icosahedral numbers up through 742 is in equivalence with 12 + 48 + 124 + 255 + 456 + 742 = 1637, which is the eighth number to reach a record amplitude between zeroes in the Mertens function (of −16).[73][183] 742 is also the thirty-eighth central polygonal number,[36] and the number of partitions of 30 into divisors of 30.[193] On the other hand, 742 is the eighth Keith number following 197,[194] which is the smallest number to reach an amplitude of 7 for M(n).[183]
  27. ^ In a toroidal board in the n–Queens problem, 132 is the count of non-attacking queens,[197] with respective indicator of 19[198] and multiplicity of 1444 = 382 [199] (where twice nineteen, is thirty-eight).[200]
  28. ^ Within other graphs, 456 is also:
    • The number of complete non-ambiguous trees of size 5.[209] These trees are defined as labelled binary trees , where each left (right, child) label appears once, and where a left (resp. right) label ancestor to is labeled strictly greater than .[210][211]
    • The number of connected regular graphs with 18 nodes and girth at least 5.[212]
    • The number of closed walks at a vertex of the complete graph on nine nodes .[213]
    • The number of degree 6 vertices in the 3-Menger sponge graph.[214]
    • The number of different compositions of the ladder graph .[215] It is also the number of (directed) Hamiltonian paths in the 6-Möbius ladder graph,[216] and the number of directed Hamiltonian paths on the 6-prism graph.[217]
    Otherwise, 456 appears in the toothpick sequence at the twenty-second generation.[218]
    Where 114 is the average of divisors of three of four numbers (302, 339, and 407) to have a sum-of-divisors of 456, 114 is also the polynomial degree of seven-cycle logistic map r7.[219][139]
  29. ^ 264 is the arithmetic mean of the sixteen divisors that belong to the fourth primary pseudoperfect number 1806 (largest of six numbers that have this value),[220][10][11] that is also the product of the first four terms (2, 3, 7, 43) in Sylvester's sequence of numbers whose reciprocals most rapidly converge to 1 (only the first four primary pseudoperfect numbers up to 1806 are products of consecutive terms in Sylvester's sequence). The aliquot sum of 1806, on the other hand, is 2418,[14] which is the seventh number after 744 where σ(φ(n)) is equal to n;[131] also, the sum-of-divisors of 264 is 720, where the sum-of-divisors of 720 is 2418.[32] The third-largest primary pseudoperfect number is 42,[220] the second-smallest sphenic number after 30,[221] whose aliquot sum it equals; 42 is specifically the reduced totient value of 1806 (that of 744 is 30, on the other hand),[26] and the composite index of 60, the smallest number with twelve divisors (12 is also the arithmetic mean of divisors of 42, and its totient value).[10][11][5] The fifth Mersenne prime after 3, 7, 31 and 127 is 8191, which is the 1028th prime number, an index value whose sum-of-divisors is 1806, with a totient of 512 = 83 [5] and a reduced totient of 256 = 28.[26]
    372 (half of 744) is the number of partitions of 42 into prime parts, where 124 is the number of such partitions of 32, and 52 (the prime index of 239, the number of totatives of 744 that are either prime or composite)[5] of 25.[17] Also, 372 is the thirtieth untouchable number (followed by 406, the index of 744 as a pernicious number), where 248 is the sixteenth (and 124 the seventh), alongside 52 the third.[150]
    1806 is also the only Bernoulli number Bn such that its denominator is also n,[222] where the first four denominators in fractions other than 1/1 that have numerators of 1 are 2, 6, 30, 42, 30 (five numbers, with 5/66 the following Bernuolli number,[223] where 66 is the third sphenic number); in this sequence 6 is also the first perfect number[81] and unitary perfect number (like 60, which is twice 30),[224] where 2 represents the first primary pseudoperfect number, before 6.
  30. ^ 71 is the twentieth prime number and 31 the eleventh. In turn, 20 is the eleventh composite number[28] that is also the sixth self-convolution of Fibonacci numbers before 38, which is the prime index of 163.
    71 is also part of the largest pair of Brown numbers (71, 7), of only three such pairs; where in its case 72 − 1 = 5040.[248][249] Consequently, both 5040 and 5041 can be represented as sums of non-consecutive factorials, following 746, 745, and 744;[2] where 5040 + 5041 = 10081 holds an aliquot sum of 611, which is the composite index of 744.[28]
    5040 is the nineteenth superabundant number[250] that is also the largest factorial that is a highly composite number,[251] and the largest of twenty-seven numbers n for which the inequality σ(n) ≥ eγnloglogn holds, where γ is the Euler–Mascheroni constant; this inequality is shown to fail for all larger numbers if and only if the Riemann hypothesis is true (known as Robin's theorem).[242] 5040 generates a sum-of-divisors 19344 = 13 × 31 × 48 that itself contains four divisors in proportion with 744 (and therefore, divisors also in proportion with 248 as well); which makes it one of only two numbers out of these twenty-seven integers n in Robin's theorem to hold σ(n) such that 744m | σ(n) for any subset of divisors m of n; the only other such number is 240:[252]
    • 19344 ÷ 26 = 744
    • 9672 ÷ 13 = 744
    • 1488 ÷ 2 = 744
    Where also 19344 ÷ 78 = 248, with 248 and 744 respectively as the 24th and 30th largest divisors (where in between these is 403 = 13 × 31, which is the middle indexed composite number congruent ±1 mod 6 less than 744 that is not part of its composite totatives); 2418, the 35th largest, is the seventh number n after 744 such that σ(φ(n)) is n.[131] Furthermore, in this sequence of integers in Robin's theorem, between 240 and 5040 lie four numbers, where the sum between the first three of these 360 + 720 + 840 = 1920 is in equivalence with σ(744). The first number to be divisible by all positive non-zero integers less than 11 is the penultimate number in this sequence 2520, where 2520 − 840 − 720 = 960 represents a Zumkeller half from the set of divisors of 744,[80] with σ(720) = 2418[14] (and while 720 + 24 = 744 = 6! + 24, where 720 is the smallest number with thirty divisors,[12] equal to 1176 − 456, a difference between the aliquot sum of 744, and the only number to have an aliquot sum of 744; wherein 720 is also in equivalence with σ(264) = 456 + 264).[14][32] 5040 = 7! = 10 × 9 × 8 × 7 is divisible by the first twelve non-zero integers, except for 11.
  31. ^ 163 is also the sum between prime indexes (81, 82) that belong to numbers adjacent to the twenty-second interprime 420 = 101 + 103 + 107 + 109, the twenty-sixth number that is the sum of four consecutive primes[253] that is also the 45th number to return 0 for the Mertens function.[92] It is the sixth such interprime to yield prime values from sums of prime indices of adjacent numbers, after (138, 72, 12, 6, 4) and preceding 432, the abundancy of 744.[24] 420, with a Euler totient of 96,[5] is half of 840, the smallest number with exactly thirty-two divisors,[12] and twice 210, the fourth primorial (2 × 3 × 5 × 7), where the previous three primorials sum to 2 + 6 + 30 = 38,[254] in equivalence with the prime index of 163.[9] Furthermore, the 163rd composite number is 210,[28] where 210 is thrice 70. Seventy is the fifth pentatope number,[255] that represents the sum of prime factors (inclusive of multiplicities) in what is described as the least-interrelated sporadic group, Janko group J3, vis-à-vis all other sporadic groups (including the pariah groups), and their relevant algebraic structures. 70 is primarily used in the construction of the Leech lattice in twenty-four dimensions, that involves the only non-trivial solution (i.e., 1) to the cannonball problem, equal to the sum of the squares of the first twenty-four integers; 70 is also equal to the number of conformal field theories in the form of vertex operator algebras with central charge of 24 with weight 1 (and aside from V0, see § E8 and the Leech lattice), that are rooted in the Leech lattice, of which the largest two exist in dimension seven hundred and forty-four. 210 is, furthermore, the first non-trivial 71-gonal number of the form P(s,n) = (s − 2)n2 − (s − 4)n/2 (for the nth polygonal number with s sides), that precedes 418, the sum of integers between 13 and 31, inclusive. Importantly, 210 is the largest number n where the number of distinct representations of n as the sum of two primes is at most the number of primes in the interval (n/2, n – 2).[256]
    420, which is the eleventh self-convolution of Fibonacci numbers,[226] preceding 744, is the totient value of 473, the first non-trivial number to hold this value (after 421). Twice 744 is 1488, which lies between the 50th pair of twin primes (1487, 1489),[8] whose successive prime indices (236, 237) add to 473. 1488 is the 235th average of successive odd primes,[59] where 235 is the sixth self-convolution of Fibonacci numbers, preceding 420,[226] with 298 the 235th composite, where 372 (half 744) is the 298th (235 is also the 183rd composite).[28] The fiftieth composite number, on the other hand, is 70.[28]
  32. ^ Worth noting, 126 is the sum of prime indices of the first class of three-digit permutable primes in decimal (113, 131, 311),[264] respectively the thirtieth, thirty-second, and sixty-fourth prime numbers[9] (where 64 is twice 32, and the 45th composite, with 45 itself the 30th composite);[28] and this sum of prime numbers is equivalent with 555. Furthermore, 126 is the seventh successive composite index starting with 13 (i.e., 13, 22, 34, 50, 70, 95, 126).[265]
    126 is equal to the sum of the composite index of 22 (13), and the twenty-second composite and prime numbers (34, 79), whose sum generated between the latter two is 113. 13 and 79 are also first members in their respective paired permutable primes in base ten (13, 31) and (79, 97), respectively the smallest and largest two-digit pairs, where 11 is the only two-digit permutable prime without a distinct dual pair with differing permuting digits.[7]
    Where 555 is generated from the first class of three-digit permutable primes in decimal, and prime and composite indices associated with 22, 131 and 126 sum to 257, which is the 55th indexed prime[9] and tenth prime number that is not a cluster prime[89] (and where, 13110 has a digit sum of 5).
    Also, 126 + 125 = 351, a value in equivalence with the twenty-sixth triangular number,[6] with 125 the cube of 5.
  33. ^ Regarding 992, the numbers 336 (the totient of 1176,[5] which is the sum-of-divisors of 744), 496 (the 31st triangular number and third perfect number),[6][81] and 525 — the sum of all prime numbers that divide the orders of the twenty-six (or twenty-seven) sporadic groups,[272] equal to the sum of the dimensions of all five exceptional Lie algebras[273] — are the first three of only four composites to have a sum-of-divisors of 992.[32] Furthermore, the fifth such number is 775, whose composite index is 637, equal to the sum of all prime factors (inclusive of multiplicities) in the order of the largest sporadic group, |1| = 246 × 320 × 59 × 76 × 112 × 133 × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71.[235]: pp.244–246  In Chowla's function (that counts the sum-of-divisors of n except for 1 and and n), 525 holds a value of 466[274] that is the prime index of 3313,[9] equal to the sum of all seven numbers (240, 350, 366, 368, 575, 671, and 743) to hold a sum-of-divisors of 744;[32] with 240 the totient of 525 as well (the 13th of 31 numbers to hold this value).[5]
  34. ^ More deeply, for the theta series of 744 is a prime-indexed coefficient over its first six indices less than 104 (respectively, the 458th, 526th, 742nd, 799th, 842nd, and 1141st prime indices, with a sum of 4058; the latter in equivalence with 1141 = 7 × 163).[282] The first composite coeff. index in the series with coefficient 744 is its seventh index 9251 = 11 × 292, whose sum of prime factors, inclusive of 1, is 70 (which is of algebraic significance in terms of the construction of the twenty-four-dimensional Leech lattice). 456, and 1176, the aliquot sum of 744,[14] are also prime-indexed coefficients over their first two coefficient indices (346th, 364th, and 1098th, 1159th, respectively).[282]
  35. ^ Where the smallest non-unitary pentagonal pyramidal number is 6, the eleventh is 726 = 6! + 6, and the twenty-fourth is 7200,[308] which is a number with a Euler totient value of 1920,[5] and a reduced totient of 120.[26]
  36. ^ Such that it can be expressed using just its digits (which are only used once, and from left to right) alongside the operators +, -, ×, ÷, ab, √, ! (with concatenation allowed).
    This, in likeness of 144 = (1 + 4)! + 4!, that is the Euler totient of 456.[5] Where the totient of 744 is 240, that of 456 is 144.[5] 187 is the composite index of 240, where 187 is the 144th composite number.[28] In turn, the sum-of-divisors of 187 is 216 = 63,[32] which is the 168th composite number.
    Also, the reduced totient of 456 is 36.[26]
  37. ^ The magic square is:

    This is the second-smallest magic constant for a 6 × 6 magic square consisting of thirty-six consecutive prime numbers, where the sum between the smallest and largest prime numbers in this square is equal to 41 + 223 = 264. The smallest such constant is 484 = 222[312] whose aliquot sum of 447 is the reverse permutation of the digits of 744 in decimal;[14] specifically, 22 and 264 are respectively the twelfth and fourteenth numbers n whose squares are undulating in decimal, while the thirteenth and penultimate such known number is 26.[313][314][315]: pp.159, 160  The smallest possible magic constant of an n × n magic square consisting only of distinct prime numbers is 120, from an n of 4,[316][317] a value equal to the arithmetic mean of all sixteen divisors of 744;[10][11] otherwise, for n = 6, the smallest magic constant for a six-by-six square with distinct prime numbers is 432,[316] also the abundance of 744.[24]
    An 11 × 11 magic square that is normal has a magic constant of 671,[318] which is the sixth and largest composite number to have a sum-of-divisors equal to 744.[32]
  38. ^ Otherwise, 992 = 248 + 744 is the least possible number of diagonals of a simple convex polyhedron with thirty-six faces; for sixteen and twenty faces, there are respectively a minimum of 132 and 240 diagonals[320] (values which represent the prime index of the largest prime totative of 744 and the count of all its totatives),[5] where 132 + 240 = 372, or half 744 (alongside respective indices that generate a sum of 36).
  39. ^ On the other hand, 456 (the only number to have an aliquot sum of 744)[14] is the maximum number of unit squares — only joined at corners — that can be inscribed inside a square (of area 1600 u2).[322]
    Adding 1 to the value of this area, 1601 = 1600 + 1, is the largest of forty lucky numbers of Euler, that is also the difference between 1804 − 203, values that represent the first and third longest sides of primitive Heronian tetrahedra with four congruent triangle faces, with the second such number equaling 144 + 744 = 888.[323][324]
Heegner numbers, finite simple groups, and lattices D4, F4, E8, Λ24
  1. ^ The smallest sporadic group is Mathieu group M11, with an irreducible complex representation in ten dimensions,[237] that is one of five first generation sporadic groups.[238] Its group order is 7920 = 8 × 9 × 10 × 11 (one less than the 103 indexed prime number),[235]: pp.244–246 [9] with a Euler totient of 1920 = σ(744).[5] M11 also holds a lowest five-dimensional faithful representation over the field with three elements, the lowest of any such group.[237] On the other hand, M12 is the second-largest Mathieu group and sporadic group by order,[235]: pp.244–246  and more specifically the thirty-first largest non-cyclic group (where M11 is the fifteenth largest),[239][240] with group order 26 × 33 × 5 × 11 = 95040 whose Euler totient (23040)[5] is divisible by 1920 twelve-times over (as its 50th largest divisor greater than 1). The nineteenth and twentieth largest non-cyclic groups are A3(2) ≃ A8 and A2(4) of orders 20160 = 26 × 32 × 5 × 7, the latter with larger outer automorphism group D12; 20160 is the twenty-third highly composite number,[241] that is divisible by both 960 (the Zumkeller half of 744),[80] and 5040 (the largest of twenty-seven numbers in Robin's theorem to hold inequality σ(n) ≥ eγnloglogn, see latter points discussing this sequence).[242] It is one less than the largest number (20161, the 1456th indexed number) which cannot be represented as the sum of two abundant numbers.[243] Otherwise, the Tits group T, the only finite simple group that can classify as of Lie type or sporadic, fits as the fifth-largest sporadic group, whose group order 211 × 33 × 52 × 13 = 17971200 holds a totient of 4423680, divisible by the sum-of-divisors of seven-hundred and forty-four 2304 times (where its 62nd largest divisor is 1728, followed by 1920).
    Of all (here, twenty-seven) sporadic groups, only the seventh largest Janko group J3 has an order, |J3| = 27 × 35 × 5 × 17 × 19 = 50232960,[235]: pp.244–246  whose totient value is not divible by 1920: 11943936 ÷ 1920 = 6022.8; where specifically its group order has a sum of prime factors (inclusive of multiplicities) equal to 70.
    Where M11 and M12 are respectively the fifteenth and thirty-first largest indexed non-cyclic groups by group order, these indices sum to 15 + 31 = 46, where the third-largest Mathieu group and fourth-largest sporadic group M22 is the forty-sixth such largest non-cyclic group.[239][240] 46 is the largest even number that cannot be represented as the sum of two abundant numbers,[243] that is also equal to the total number of maximal subgroups of the Friendly Giant ,[244] which collectively holds 20 of 26 sporadic groups as subquotients (strictly).[235]: pp.244–246 
    46 is the thirty-first composite,[28] whose aliquot sum is 26, the only number to hold this value.[14]
  2. ^ The sequence of self-convoluted Fibonacci numbers starts {0, 0, 1, 2, 5, 10, 20, 38, 71, 130, 235, 420, 744, 1308, 2285, 3970, 6865, 11822...}.[226] The sum of the first seven terms (from the zeroth through the sixth term) is equal to 38, which is equivalent to the a(n) = 7 member in this sequence. Taking the sum of the three terms that lie between 71[xxx] and 744 (i.e. 130, 235, 420) yields 785 = 282 + 1, whose aliquot sum is 163,[14][xxxi] the thirty-eighth prime number.[9] 785 is the 60th number to return 0 for the Mertens function, which also includes 163, the 13th such number.[92] 785 is also the number of irreducible planted trees (of root vertex having degree one) with six leaves of two colors.[257]
    1106, the first member in the sum of consecutive composite numbers (in equivalence with 2229 = 743 × 3) that are divisible by the twenty-fifth prime number 97,[258] is the smallest number to return a value of 14 in the inverse Mertens function,[183] where 1106 lies between the 81st pair of sexy primes (1103, 1109).[259] 14 is the floor (and nearest integer)[260] of the imaginary part of the first non-trivial zero in the Riemann zeta function ζ, and the fourteenth indexed floor value is 60[261][262] — sixty is the smallest number with exactly twelve divisors, and there are only two numbers which have a sum-of-divisors of 60: 24 and 38,[32] whose sum is 62 (which is the tenth largest divisor of 744); also, three of nine numbers with totients of 60 are also divisors of 744 (93, 124, and 186, that add to 403).[5] On the other hand, the difference 38 − 24 = 14, where the sixtieth floor value in the imaginary part of its nontrivial zero in ζ is the largest Heegner number 163.
  3. ^ 163 is the thirty-eighth indexed prime and 67 the nineteenth (with the biprime 38 = 2 × 19), for values d. In the approximation for the almost integer containing the fourteenth prime number as the seventh Heegner number 43 for d, 960 is the smallest number with exactly twenty-eight divisors,[12] equivalent to the sum of either of two sets of divisors of 744 that collectively add to 1920, as mentioned in § Totients. Likewise, in the approximation for the almost integer containing the eighth Heegner number 67 for d, 5280 is equal to the sum between 240 and 5040, which are the only two numbers in the set of twenty-seven integers in Robin's theorem for the Riemann hypothesis that have a set of divisors whose sums are divisible by 744;[252] 5280 also lies between the 131st pair of twin primes (5281, 5279),[8] respectively the 701st and 700th prime numbers, where 5281 is the 126th super-prime.[61][xxxii]
  4. ^ In the list of thirteen integers n ≤ 1000 that yield almost integers — with nearness |nint(x) − x| ≤ 10−3 for values x ≡ eπ√n — the smallest n of these is 25, that is also the smallest composite totative of 744. Less than 103, the largest such n is 719 = 6! − 1, where 719 represents the 128th indexed prime number;[9] the sum generated by these two n is in equivalence with 719 + 25 = 744.
    The middle indexed value in between these two bounds is 148, the twelfth square-free positive integer d over the negated imaginary quadratic field of class number 2, followed by 163 and 232, the latter of which is the fourteenth square-free positive integer d over the imaginary quadratic field √−d of class number 2 (only these two numbers 148 and 232 for d in this field of class number 2 yield almost integers with |nint(x) − x| ≤ 10−3, where 163 is the largest over the same field with class number 1).[267] In this same list, the three distinct non-supersingular primes (37, 43, 67) that only divide orders of pariah groups are also n (less than 148) that yield almost integers of the form eπ√n. In the almost integer representation for eπ√148 specified, 8 × 103 = j(√(2i)) = 203, the class 1 j−invariant of order d = −8,[268] that is also a root in Ψ2(X, X) = −(X − 8000) × (X + 3375)2 × (X − 1728), a polynomial for supersingular 2-isogeny graphs with loops (it is also a root in Ψ3).[269][270] The remaining two n (58, 74) in the list of these almost integers with nearness |nint(x) − x| ≤ 10−3 of n less than 148 generate a sum of 58 + 74 = 132 equal to the prime index of 743,[9] the largest prime totative of 744 (and, where 2 × 37 = 74 = 148 ÷ 2).
  5. ^ For almost integers with nearness |nint(x) − x| ≤ 10−2, the smallest such number is eπ√6 with n of 6, approximately equal to 2197.99087.[271] Whereas 2198 holds eight divisors that produce an arithmetic mean of 474[10][11] (and where 474 holds a sum-of-divisors equal to 960,[32] also the Zumkeller half from the set of divisors of 744),[80] 2199 is the sixteenth perfect totient number,[34] with an aliquot sum of 737[14] equivalent with the largest composite totative of 744. Also, 2197 = 133.
    There are a total of twenty-six almost integers with |nint(x) − x| ≤ 10−2 where x ≡ eπ√n and n ≤ 103, where the largest n holds a value of 986.[271] The sum between the upper and lower bounds n less than one thousand with this almost integer degree of nearness is equal to 6 + 986 = 992 = 744 + 248.[xxxiii]
    Otherwise, the difference between the largest class 1 and 2 square-free integers is 427 − 163 = 264. Regarding square-free integers of class number h = 3, there are a total of sixteen (or twenty-five when including non-maximal orders),[275] the largest with value of 907 that is the 155th indexed prime number,[9] equal to the sum 163 + 744, that is also the greater of consecutive primes (887, 907) that generate the seventh largest record prime gap (of 20).[75][76][77][78][79] The previous greatest prime gap is of 18 set by (523, 541) (following the thirtieth and thirty-first prime numbers 113 and 127 with a gap of 14), the 99th and 100th prime numbers,[9] the former also the composie index of 132,[28] and the latter the tenth star number[146] and 53rd number to return 0 for the Mertens function (where 427 is the 50th, and 163 the 13th).[92] Of the square-free integers of class number three, only 59 produces an almost integer of the form x ≡ eπ√n with |nint(x) − x| ≤ 10−2 (for integers of class number h = 4, the largest such number is 1555 — whose square divides 61554[276] — while the ninth such largest discriminant is 155).[277] In the full list of largest square-free positive integers with class numbers h ≤ 100,[278][279] ten of these maximum values are held by larger integers when non-maximal orders in their respective classes are included, where nine of these are uniquely biprimes divisible by 163 (the only exception is the largest value for square-free integers of class number 16); the largest of these in this bound is 821683 = 712 × 163 of class number h = 70.[263]: 18−20 
  6. ^ ‍+
    4     ≅     as a lattice is the union of two lattices,[281] where the associated theta series of has 744 as its 50th indexed coefficient (as with theta series of ),[284][280] and where its twenty-fifth coefficient is 248, which is the most important divisor of 744. Also, in this theta series of , the preceding 49th coefficient is 456, the only number to hold an aliquot sum of 744,[14] where 744 − 456 = 288, a value that is the number of cells in the disphenoidal 288-cell, whose 48 vertices collectively represent the twenty-four Hurwitz unit quaternions with norm squared 1, united with the twenty-four vertices of the dual 24-cell with norm squared 2. For the theta series of , that is realized in the 16-cell honeycomb, all 25 × 2m indexed coefficients (i.e. 25, 50, 100, 200, 400, ...) are 744.[284] For both the theta series of and , the 456th coefficient is 1920, the sum-of-divisors of 744.[32]
    Also, for the theta series of the four-dimensional lattice, coefficient index 288 is the 97th non-zero norm, coeff. index 456 the 153rd (an index that represents the seventeenth triangular number), and coeff. index 744 the 250th;[283] the latter, a coefficient index that is the largest of only four numbers to hold a Euler totient of 100: 101, 125, 202, and 250[5] — the smallest of these is the twenty-sixth prime number, while the largest 250 = 2 × 53 has a sum of prime factors that is 17, the seventh prime number;[9] and where 202 = 49 + 153 and 250 = 153 + 97, with 549 = 49 + 97 + 153 + 250, the 447th indexed composite number.[28][xxxiv]
  7. ^ The Petrie polygon of the latter two which are rooted in icosahedral symmetry is the thirty-sided triacontagon, while the former has a dodecagon as its like-polygon, whose central angle is of thirty degrees (and internal angle of 150 degrees). The twenty-fourth prime is 89,[9] which is the index of 120 in the list of composites.[28] In a 600-cell fit precisely seventy-five 16-cells and seventy-five tesseracts, where there also fit twenty-five 24-cells. Where twenty-five 24-cells also fit inside a 24-cell honeycomb as a unit (with twenty-four 24-cells surrounding a single 24-cell), whose symmetry is rooted in , a different arrangement of twenty-five 24-cells fits in a 600-cell with symmetry. , in eight-dimensional space, has a Coxeter number of 30, and a symmetry that traces back to the 600-cell via the icosians.
  8. ^ The 8 lattice holds two hundred and forty root vectors that are represented by the vertex arrangement of the 421 polytope, whose Petrie polygon is a thirty-sided triacontagon,[293][294] where 30 is the Coxeter number h of Coxeter group E8. Also, Coxeter numbers of simple reflexions in E6 and E7 are of orders 12 and 18, together equivalent to the Coxeter number of E8;[285]: 234  where E6 and E7 embed inside E8. Four-dimensional H4 hexadecachoric symmetry also contains a Coxeter number of 30, where H4 is the higher-dimensional analogue of icosahedral symmetry Ih.
    This 8 lattice structure with 240 root vectors can be constructed with 120 quaternionic unit icosians that form the vertices of the four-dimensional 600-cell, whose symmetries are rooted in three-dimensional icosahedral symmetry Ih of order 120,[295] a value equal to the total number of reflections of Coxeter group E8.[285]: 226–232  In total, a regular icosahedron and dodecahedron contain thirty-one axes of symmetry; six five-fold, ten three-fold, and fifteen two-fold.[296]
    Where the arithmetic mean of divisors of 744 is 120,[10][11] with a largest prime factor of 31 and a reduced totient λ(n) of 30,[26] the number of integers that are relatively prime with and up to 744 is 240,[5] with the sum-of-divisors σ(240) = 744.[287]: p.21 
  9. ^ VE‍3
    8         is isomorphic to the tensor product VE8 ⊗ VE8 ⊗ VE8 ; also, affine structure D16 is different from D‍+
    16    , which is associated with even positive definite unimodular lattice ‍+
    16    .[300][301]
    E‍3
    8,1     and D16,1E8,1 are associated with codes e‍3
    8     and d16e8 that are two of only nine in-equivalent doubly even self-dual codes of length 24 and weight 4.[302][303] The largest of these VOAs VD24 is realized in dim 1128, where successively halving its dimensional space leads to a 70+1⁄2–dimensional space.

References

  1. ^ a b Sloane, N. J. A. (ed.). "Sequence A189975 (Numbers with prime factorization pqr^3.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  2. ^ a b Sloane, N. J. A. (ed.). "Sequence A060112 (Sums of nonconsecutive factorial numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A083539 (a(n) is sigma(n) * sigma(n+1) as the product of sigma-values for consecutive integers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  5. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  6. ^ a b c d e f g h i j k l m n Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular number: a(n) is the binomial(n+1,2) equivalent to n*(n+1)/2 that is 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  7. ^ a b c Sloane, N. J. A. (ed.). "Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-14.
  8. ^ a b c d e f g Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.
  9. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
  10. ^ a b c d e f g h i j k l m n o p q r s t u v Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  11. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  12. ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-06.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A000005 (d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-28.
  14. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  15. ^ Sloane, N. J. A. (ed.). "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-09-22.
  16. ^ Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-22.
  17. ^ a b c Sloane, N. J. A. (ed.). "Sequence A000607 (Number of partitions of n into prime parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  18. ^ Sloane, N. J. A. (ed.). "Sequence A347586 (Number of partitions of n into at most 4 distinct parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  19. ^ a b Sloane, N. J. A. (ed.). "Sequence A007947 (Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-31.
  20. ^ a b Sloane, N. J. A. (ed.). "Sequence A005277 (Nontotients: even numbers k such that phi(m) equal to k has no solution.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
  21. ^ Sloane, N. J. A. (ed.). "Sequence A005278 (Noncototients: numbers k such that x - phi(x) equal to k has no solution.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  22. ^ Sloane, N. J. A. (ed.). "Sequence A001207 (Number of fixed hexagonal polyominoes with n cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  23. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
  24. ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A033880 (Abundance of n, or (sum of divisors of n) - 2n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-29.
  25. ^ a b c d e f g h i j k l Sloane, N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  26. ^ a b c d e f g h i Sloane, N. J. A. (ed.). "Sequence A002322 (Reduced totient function psi(n): least k such that x^k is congruent 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
  27. ^ a b c Sloane, N. J. A. (ed.). "Sequence A014574 (Average of twin prime pairs.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  28. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-27.
  29. ^ Sloane, N. J. A. (ed.). "Sequence A217843 (Numbers which are the sum of one or more consecutive nonnegative cubes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  30. ^ Sloane, N. J. A. (ed.). "Sequence A001694 (Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  31. ^ Sloane, N. J. A. (ed.). "Sequence A052486 (Achilles numbers - powerful but imperfect: if n equal to Product(p_i^e_i) then all e_i less than 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-29.
  32. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z Sloane, N. J. A. (ed.). "Sequence A000203 (a(n) equal to sigma(n), the sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
  33. ^ Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
  34. ^ a b Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-21.
  35. ^ Sloane, N. J. A. (ed.). "Sequence A006531 (Semiorders on n elements.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-02.
  36. ^ a b Sloane, N. J. A. (ed.). "Sequence A002061 (Central polygonal numbers: a(n) equal to n^2 - n + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-02.
  37. ^ Sloane, N. J. A. (ed.). "Sequence A139250 (Toothpick sequence (see Comments lines for definition).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-02.
  38. ^ Sloane, N. J. A. (ed.). "Sequence A003136 (Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-02.
  39. ^ Mitchell, David (February 25, 2017). "Tessellating the Ramanujan-Hardy Taxicab Number, 1729, Bedrock of Integer Sequence A198775". LatticeLabyrinths. Retrieved 19 July 2018.
  40. ^ Sloane, N. J. A. (ed.). "Sequence A198775 (Numbers having exactly four representations by the quadratic form x^2+xy+y^2 with 0 less than or equal to x less than or equal to y.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-03.
  41. ^ Sloane, N. J. A. (ed.). "Sequence A198773 (Numbers having exactly two representations by the quadratic form x^2+xy+y^2 with 0 less than or equal to x less than or equal to y.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-03.
  42. ^ Sloane, N. J. A. (ed.). "Sequence A118886 (Numbers expressible as x^2 + x*y + y^2, 0 less than or equal to x less than or equal to y in 2 or more ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-03.,
  43. ^ Marshall, John U. (1975). "The Loeschian numbers as a problem in number theory" (PDF). Geographical Analysis. 7 (4): 421–426. Bibcode:1975GeoAn...7..421M. doi:10.1111/j.1538-4632.1975.tb01054.x – via OEIS.
  44. ^ Sloane, N. J. A. (ed.). "Sequence A001107 (10-gonal (or decagonal) numbers: a(n) is n*(4*n-3).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-03.
  45. ^ a b Sloane, N. J. A. (ed.). "Sequence A090785 (Numbers that can be expressed as the difference of the squares of consecutive primes in just one way.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-03.
  46. ^ a b Sloane, N. J. A. (ed.). "Sequence A010785 (Repdigit numbers, or numbers whose digits are all equal.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-19.
  47. ^ Sloane, N. J. A. (ed.). "Sequence A000048 (Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-03.
  48. ^ Diaconis, Persi; Graham, Ron (2011). "Chapter 5: From the Gilbreath Principle to the Mandelbrot Set" (PDF). Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks. Princeton University Press. pp. 61–83. doi:10.1515/9781400839384-007. ISBN 978-1-4008-3938-4. MR 2858033..
  49. ^ Sloane, N. J. A. (ed.). "Sequence A090788 (Numbers that can be expressed as the difference of the squares of primes in just two distinct ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-03.
  50. ^ a b Sloane, N. J. A. (ed.). "Sequence A005153 (Practical numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
  51. ^ Sloane, N. J. A. (ed.). "Sequence A345547 (Numbers that are the sum of nine cubes in eight or more ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  52. ^ Sloane, N. J. A. (ed.). "Sequence A075308 (Number of n-digit perfect powers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  53. ^ a b c Sloane, N. J. A. (ed.). "Sequence A029954 (Palindromic in base 7.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  54. ^ Sloane, N. J. A. (ed.). "Sequence A052294 (Pernicious numbers: numbers with a prime number of 1's in their binary expansion.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-28.
  55. ^ Sloane, N. J. A. (ed.). "Sequence A078392 (Sum of GCD's of parts in all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
  56. ^ Sloane, N. J. A. (ed.). "Sequence A064986 (Number of partitions of n into factorial parts (0! not allowed).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
  57. ^ Sloane, N. J. A. (ed.). "Sequence A316433 (Number of integer partitions of n whose length is equal to the LCM of all parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
  58. ^ Sloane, N. J. A. (ed.). "Sequence A000166 (Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
  59. ^ a b c Sloane, N. J. A. (ed.). "Sequence A024675 (Average of two consecutive odd primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  60. ^ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem: g(n) equal to 2^n + floor((3/2)^n) - 2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
  61. ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.
  62. ^ a b Andrews, William Symes (1917). Magic Squares and Cubes (PDF). Chicago, IL: Open Court Publishing Company. pp. 176, 177. ISBN 9780486206585. MR 0114763. OCLC 1136401. Zbl 1003.05500.
  63. ^ Sloane, N. J. A. (ed.). "Sequence A021023 (Decimal expansion of 1/19.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  64. ^ Sloane, N. J. A. (ed.). "Sequence A072359 (Primes p such that the p-1 digits of the decimal expansion of k/p (for k equal to 1,2,3,...,p-1) fit into the k-th row of a magic square grid of order p-1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  65. ^ Sloane, N. J. A. (ed.). "Sequence A015723 (Number of parts in all partitions of n into distinct parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-20.
  66. ^ a b Sloane, N. J. A. (ed.). "Sequence A034885 (Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  67. ^ Sloane, N. J. A. (ed.). "Sequence A006002 (a(n) equal to n*(n+1)^2/2 (Sum of the nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ...))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
  68. ^ Sloane, N. J. A. (ed.). "Sequence A081377 (Numbers n such that the set of prime divisors of phi(n) is equal to the set of prime divisors of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  69. ^ Sloane, N. J. A. (ed.). "Sequence A002822 (Numbers m such that 6m-1, 6m+1 are twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.
  70. ^ Sloane, N. J. A. (ed.). "Sequence A161680 (a(n) is the binomial(n,2): number of size-2 subsets of {0,1,...,n} that contain no consecutive integers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.
  71. ^ Sloane, N. J. A. (ed.). "Sequence A000931 (Padovan sequence (or Padovan numbers)...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-02.
  72. ^ Sloane, N. J. A. (ed.). "Sequence A007304 (Sphenic numbers: products of 3 distinct primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-08.
  73. ^ a b Sloane, N. J. A. (ed.). "Sequence A304241 (Indices where Mertens function A002321 reaches record amplitudes between zeros.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-08.
  74. ^ Sloane, N. J. A. (ed.). "Sequence A002321 (Mertens's function: Sum_{k equal to 1..n} mu(k), where mu is the Moebius function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-08.
  75. ^ a b c Sloane, N. J. A. (ed.). "Sequence A005250 (Record gaps between primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-11.
  76. ^ a b c Sloane, N. J. A. (ed.). "Sequence A005669 (Indices of primes where largest gap occurs.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-11.
  77. ^ a b c Sloane, N. J. A. (ed.). "Sequence A001223 (Prime gaps: differences between consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-11.
  78. ^ a b c Sloane, N. J. A. (ed.). "Sequence A002386 (Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-11.
  79. ^ a b c Sloane, N. J. A. (ed.). "Sequence A000101 (Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-11.
  80. ^ a b c d Sloane, N. J. A. (ed.). "Sequence A083207 (Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  81. ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  82. ^ Sloane, N. J. A. (ed.). "Sequence A000372 (Dedekind numbers or Dedekind's problem: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  83. ^ Sloane, N. J. A. (ed.). "Sequence A006880 (Number of primes less than 10^n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  84. ^ Sloane, N. J. A. (ed.). "Sequence A001013 (Jordan-Polya numbers: products of factorial numbers A000142.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-12.
  85. ^ Sloane, N. J. A. (ed.). "Sequence A127333 (Numbers that are the sum of 6 consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
  86. ^ Sloane, N. J. A. (ed.). "Sequence A000009 (Expansion of Product_{m greater than or equal to 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  87. ^ Sloane, N. J. A. (ed.). "Sequence A228486 (Near primorial primes: primes p such that p+1 or p-1 is a primorial number (A002110).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-16.
  88. ^ Sloane, N. J. A. (ed.). "Sequence A038509 (Composite numbers congruent to +-1 mod 6.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  89. ^ a b Sloane, N. J. A. (ed.). "Sequence A038133 (From a subtractive Goldbach conjecture: odd primes that are not cluster primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.
  90. ^ Blecksmith, Richard; Erdős, Paul; Selfridge, J. L. (1999). "Cluster Primes". The American Mathematical Monthly. 106 (1): 43. doi:10.2307/2589585. JSTOR 2589585. Zbl 0985.11041.
  91. ^ Sloane, N. J. A. (ed.). "Sequence A002476 (Primes of the form 6m + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.
  92. ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  93. ^ Sloane, N. J. A. (ed.). "Sequence A034961 (Sums of three consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  94. ^ Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  95. ^ Sloane, N. J. A. (ed.). "Sequence A056107 (Third spoke of a hexagonal spiral.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  96. ^ Sloane, N. J. A. (ed.). "Sequence A257750 (Quasi-Carmichael numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  97. ^ Sloane, N. J. A. (ed.). "Sequence A092269 (Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  98. ^ Sloane, N. J. A. (ed.). "Sequence A045931 (Number of partitions of n with equal number of even and odd parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  99. ^ Sloane, N. J. A. (ed.). "Sequence A000096 (a(n) equal to n*(n+3)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  100. ^ a b Sloane, N. J. A. (ed.). "Sequence A006883 (Long period primes: the decimal expansion of 1/p has period p-1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  101. ^ Sloane, N. J. A. (ed.). "Sequence A001567 (Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  102. ^ Sloane, N. J. A. (ed.). "Sequence A005898 (Centered cube numbers: n^3 + (n+1)^3.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  103. ^ Sloane, N. J. A. (ed.). "Sequence A047966 (a(n) equal to Sum_{ d divides n } q(d), where q(d) is A000009 equal to number of partitions of d into distinct parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  104. ^ Sloane, N. J. A. (ed.). "Sequence A000695 (Moser-de Bruijn sequence: sums of distinct powers of 4.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  105. ^ Sloane, N. J. A. (ed.). "Sequence A053699 (a(n) is n^4 + n^3 + n^2 + n + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  106. ^ Sloane, N. J. A. (ed.). "Sequence A000567 (Octagonal numbers: n*(3*n-2). Also called star numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  107. ^ Sloane, N. J. A. (ed.). "Sequence A007569 (Number of nodes in regular n-gon with all diagonals drawn.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  108. ^ Sloane, N. J. A. (ed.). "Sequence A001045 (Jacobsthal sequence (or Jacobsthal numbers)...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  109. ^ Sloane, N. J. A. (ed.). "Sequence A046746 (Sum of smallest parts of all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-28.
  110. ^ Sloane, N. J. A. (ed.). "Sequence A003215 (Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  111. ^ Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers: a(n) equal to n*(5*n-4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  112. ^ Sloane, N. J. A. (ed.). "Sequence A195316 (Centered 36-gonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  113. ^ Sloane, N. J. A. (ed.). "Sequence A005936 (Pseudoprimes to base 5.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  114. ^ Sloane, N. J. A. (ed.). "Sequence A016105 (Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  115. ^ Sloane, N. J. A. (ed.). "Sequence A055233 (Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  116. ^ Sloane, N. J. A. (ed.). "Sequence A000019 (Number of primitive permutation groups of degree n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  117. ^ Sloane, N. J. A. (ed.). "Sequence A002817 (Doubly triangular numbers: a(n) equal to n*(n+1)*(n^2+n+2)/8.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-24.
  118. ^ Sloane, N. J. A. (ed.). "Sequence A202018 (a(n) equal to n^2 + n + 41.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  119. ^ Sloane, N. J. A. (ed.). "Sequence A005846 (Primes of the form n^2 + n + 41.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  120. ^ Sloane, N. J. A. (ed.). "Sequence A152658 (Beginnings of maximal chains of primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  121. ^ Sloane, N. J. A. (ed.). "Sequence A033664 (Every suffix is prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  122. ^ Sloane, N. J. A. (ed.). "Sequence A066967 (Total sum of odd parts in all partitions of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  123. ^ Sloane, N. J. A. (ed.). "Sequence A007279 (Number of partitions of n into partition numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  124. ^ Sloane, N. J. A. (ed.). "Sequence A032032 (Number of ways to partition n labeled elements into sets of sizes of at least 2 and order the sets.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  125. ^ Sloane, N. J. A. (ed.). "Sequence A343377 (Number of strict integer partitions of n with no part divisible by all the others.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  126. ^ Sloane, N. J. A. (ed.). "Sequence A285900 (Sum of all parts of all partitions of all positive integers less than or equal to n into consecutive parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  127. ^ a b c Sloane, N. J. A. (ed.). "Sequence A319274 (Osiris or Digit re-assembly numbers: numbers that are equal to the sum of permutations of subsamples of their own digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  128. ^ Sloane, N. J. A. (ed.). "Sequence A088825 (Numbers n such that the sum of largest prime factors of numbers from 1 to n is divisible by n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-25.
  129. ^ Sloane, N. J. A. (ed.). "Sequence A321719 (Number of non-normal semi-magic squares with sum of entries equal to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.
  130. ^ Sloane, N. J. A. (ed.). "Sequence A020994 (Primes that are both left-truncatable and right-truncatable.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-13..
  131. ^ a b c Sloane, N. J. A. (ed.). "Sequence A018784 (Numbers n such that sigma(phi(n)) is n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  132. ^ Sloane, N. J. A. (ed.). "Sequence A017765 (Binomial coefficients C(49,n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  133. ^ Sloane, N. J. A. (ed.). "Sequence A029600 (Numbers in the (2,3)-Pascal triangle (by row).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
    The row is {2, 23, 120, 375, 780, 1134, 1176, 870, 450, 155, 32, 3}.
  134. ^ Sloane, N. J. A. (ed.). "Sequence A001263 (Triangle of Narayana numbers T(n,k) is C(n-1,k-1)*C(n,k-1)/k with 1 less than or equal to k less than or equal to n, read by rows. Also called the Catalan triangle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  135. ^ Sloane, N. J. A. (ed.). "Sequence A105278 (Triangle read by rows: T(n,k) equal to binomial(n,k)*(n-1)!/(k-1)!.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  136. ^ Sloane, N. J. A. (ed.). "Sequence A127787 (Numbers n such that F(n) divides F(F(n)), where F(n) is a Fibonacci number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  137. ^ Sloane, N. J. A. (ed.). "Sequence A046306 (Numbers that are divisible by exactly 6 primes with multiplicity.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  138. ^ a b c Sloane, N. J. A. (ed.). "Sequence A002620 (Quarter-squares: a(n) equal to floor(n/2)*ceiling(n/2). Equivalently, a(n) is floor(n^2/4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-30.
  139. ^ a b Sloane, N. J. A. (ed.). "Sequence A118454 (Algebraic degree of the onset of the logistic map n-bifurcation.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-29.
  140. ^ Bailey, D. H.; Borwein, J. M.; Kapoor, V.; Weisstein, E. W. (2006). "Ten Problems in Experimental Mathematics" (PDF). American Mathematical Monthly. 113 (6). Taylor & Francis: 482–485. doi:10.2307/27641975. JSTOR 27641975. MR 2231135. S2CID 13560576. Zbl 1153.65301 – via JSTOR.
  141. ^ Sloane, N. J. A. (ed.). "Sequence A091517 (Decimal expansion of the value of r corresponding to the onset of the period 16-cycle in the logistic map.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-29.
  142. ^ Sloane, N. J. A. (ed.). "Sequence A006562 (Balanced primes (of order one): primes which are the average of the previous prime and the following prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-06.
  143. ^ Sloane, N. J. A. (ed.). "Sequence A046119 (Middle member of a sexy prime triple: value of p+6 such that p, p+6 and p+12 are all prime, but p+18 is not (although p-6 might be).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-06.
  144. ^ Sloane, N. J. A. (ed.). "Sequence A046117 (Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-06.
  145. ^ Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime (Lesser of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-06.
  146. ^ a b Sloane, N. J. A. (ed.). "Sequence A003154 (Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-06.
  147. ^ Sloane, N. J. A. (ed.). "Sequence A083577 (Prime star numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-06.
  148. ^ a b Sloane, N. J. A. (ed.). "Sequence A006564 (Icosahedral numbers: a(n) equal to n*(5*n^2 - 5*n + 2)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-23.
  149. ^ a b Sloane, N. J. A. (ed.). "Sequence A007588 (Stella octangula numbers: a(n) equal to n*(2*n^2 - 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-17.
  150. ^ a b c Sloane, N. J. A. (ed.). "Sequence A005114 (Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-17.
  151. ^ Sloane, N. J. A. (ed.). "Sequence A051177 (Perfectly partitioned numbers: numbers k that divide the number of partitions p(k).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-23.
  152. ^ Sloane, N. J. A. (ed.). "Sequence A005432 (Number of permutation groups of degree n (or, number of distinct subgroups of symmetric group S_n, counting conjugates as distinct).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  153. ^ Sloane, N. J. A. (ed.). "Sequence A067659 (Number of partitions of n into distinct parts such that number of parts is odd.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  154. ^ Sloane, N. J. A. (ed.). "Sequence A067661 (Number of partitions of n into distinct parts such that number of parts is even.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  155. ^ Sloane, N. J. A. (ed.). "Sequence A127337 (Numbers that are the sum of 10 consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-16.
  156. ^ a b c Beeler, M.; Gosper, R.W.; Schroeppel, R. (1972-02-29). HAKMEM (PDF) (Report). MIT Computer Science and Artificial Intelligence Laboratory. p. 24 (ITEM 63). AIM 239. Retrieved 2023-12-19 – via Princeton Plasma Physics Laboratory.
    "The joys of 239 are as follows:
    pi = 16 arctan (1/5) − 4 arctan(1/239),
    which is related to the fact that 2 * 134 − 1 = 2392,
    which is why 239/169 is an approximant (the 7th) of √2.
    arctan(1/239) = arctan(1/70) − arctan(1/99)
                           = arctan(1/408) + arctan(1/577)
    239 needs 4 squares (the maximum) to express it.
    239 needs 9 cubes (the maximum, shared only with 23) to express it.
    239 needs 19 fourth powers (the maximum) to express it.
    (Although 239 doesn't need the maximum number of fifth powers.)
    1/239 = .00418410041841..., which is related to the fact that
    1,111,111 = 239 * 4,649.
    The 239th Mersenne number, 2239 − 1, is known composite,
              but no factors are known.
    239 = 11101111 base 2.
    239 = 22212 base 3.
    239 = 3233 base 4.
    There are 239 primes < 1500.
    K239 is Mozart's only work for 2 orchestras.
    Guess what memo this is.
    And 239 is prime, of course."
  157. ^ Sloane, N. J. A. (ed.). "Sequence A078125 (Number of partitions of 3^n into powers of 3)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  158. ^ Siksek, Samir (1995). Descents on Curves of Genus I (PDF) (Ph.D. thesis). Exeter, UK: University of Exeter. pp. 16–17. hdl:10871/8323. S2CID 118846642.
  159. ^ Sloane, N. J. A. (ed.). "Sequence A229384 (Positive integer solutions y1, x1, y2, x2 to Ljunggren's equation x^2 + 1 equal to 2y^4.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-19.
  160. ^ Sloane, N. J. A. (ed.). "Sequence A001333 (Numerators of continued fraction convergents to sqrt(2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-19.
  161. ^ Sloane, N. J. A. (ed.). "Sequence A000110 (Bell or exponential numbers: number of ways to partition a set of n labeled elements.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-19.
  162. ^ Sloane, N. J. A. (ed.). "Sequence A074902 (Known friendly numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
  163. ^ Sloane, N. J. A. (ed.). "Sequence A072938 (Highly composite numbers that are half of the next highly composite number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-19.
  164. ^ Sloane, N. J. A. (ed.). "Sequence A350028 (Number of Euler tours of the complete graph on n vertices (minus a matching if n is even).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  165. ^ Sloane, N. J. A. (ed.). "Sequence A232545 (Number of Euler tours of the complete digraph on n vertices.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  166. ^ Sloane, N. J. A. (ed.). "Sequence A027683 (Number of independent vertex sets of the n X n torus grid graph.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  167. ^ Sloane, N. J. A. (ed.). "Sequence A027624 (Number of independent vertex sets in the n-hypercube graph Q_n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  168. ^ Sloane, N. J. A. (ed.). "Sequence A198303 (Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth exactly g.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  169. ^ Sloane, N. J. A. (ed.). "Sequence A186734 (Triangular array C(n,k) counting connected k-regular simple graphs on n vertices with girth exactly 4.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-15.
  170. ^ Sloane, N. J. A. (ed.). "Sequence A006924 (Number of connected trivalent graphs with 2n nodes and girth exactly 4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  171. ^ Sloane, N. J. A. (ed.). "Sequence A098623 (Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  172. ^ Gilbert, Labelle (2000). "Counting enriched multigraphs according to the number of their edges (or arcs)". Discrete Mathematics. 217 (1–3). Amsterdam: Elsevier: 237–248. doi:10.1016/S0012-365X(99)00265-4. MR 1766269. S2CID 22434588. Zbl 0959.05061.
  173. ^ Sloane, N. J. A. (ed.). "Sequence A022801 (n-th Lucas number (A000204(n)) + n-th non-Lucas number (A090946(n+1)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  174. ^ Sloane, N. J. A. (ed.). "Sequence A000204 (Lucas numbers (beginning with 1): L(n) equal to L(n-1) + L(n-2) with L(1) equal to 1, L(2) equal to 3.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  175. ^ Sloane, N. J. A. (ed.). "Sequence A090946 (Non-Lucas numbers: complement of A000204.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  176. ^ Sloane, N. J. A. (ed.). "Sequence A114312 (Number of partitions of n with distinct numbers of odd and even parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  177. ^ Sloane, N. J. A. (ed.). "Sequence A171967 (Number of partitions of n with distinct numbers of odd and even parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  178. ^ Sloane, N. J. A. (ed.). "Sequence A253927 (Number of polyhexes (without holes) of size 6*n+1 with 6-fold rotational symmetry.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  179. ^ Sloane, N. J. A. (ed.). "Sequence A212789 (Number of endofunctions on [n] with distinct cycle lengths.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  180. ^ Sloane, N. J. A. (ed.). "Sequence A074127 (Group the composite numbers so that the sum of the n-th group is a multiple of the n-th prime: (4), (6), (8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22), (24, 25), (26, 27, 28, 30, 32), (33, 34, ...), ... Sequence gives the group sum divided by n-th prime for the n-th group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  181. ^ Sloane, N. J. A. (ed.). "Sequence A060354 (The n-th n-gonal number: a(n) equal to n*(n^2 - 3*n + 4)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-22.
  182. ^ Sloane, N. J. A. (ed.). "Sequence A051866 (14-gonal (or tetradecagonal) numbers: a(n) equal to n*(6*n-5).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-22.
  183. ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A051402 (Inverse Mertens function: smallest k such that |M(k)| is equal to n, where M(x) is Mertens's function A002321.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-22.
  184. ^ Sloane, N. J. A. (ed.). "Sequence A051890 (a(n) equal to 2*(n^2 - n + 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-22.
  185. ^ Sloane, N. J. A. (ed.). "Sequence A001111 (Number of inequivalent Hadamard designs of order 4n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-22.
  186. ^ Spence, Edward. "Hadamard matrices". Edward Spence (Homepage). University of Glasgow Department of Mathematics. Retrieved 2024-01-22.
  187. ^ a b Sloane, N. J. A. (ed.). "Sequence A304241 (Indices where Mertens function A002321 reaches record amplitudes between zeros.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-22.
  188. ^ Sloane, N. J. A. (ed.). "Sequence A001414 (Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-18.
  189. ^ Sloane, N. J. A. (ed.). "Sequence A046197 (Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-23.
  190. ^ Sloane, N. J. A. (ed.). "Sequence A005188 (Armstrong (or pluperfect, or Plus Perfect, or narcissistic) numbers: m-digit positive numbers equal to sum of the m-th powers of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-23.
  191. ^ a b Sloane, N. J. A. (ed.). "Sequence A006145 (Ruth-Aaron numbers (1): sum of prime divisors of n equal to the sum of prime divisors of n+1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-23.
  192. ^ Sloane, N. J. A. (ed.). "Sequence A006146 (Sums of prime divisors of Ruth-Aaron numbers (A006145).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-23.
  193. ^ Sloane, N. J. A. (ed.). "Sequence A018818 (Number of partitions of n into divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-07.
  194. ^ Sloane, N. J. A. (ed.). "Sequence A007629 (Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-07.
  195. ^ Sloane, N. J. A. (ed.). "Sequence A357886 (Triangle read by rows: T(n,k) is the number of open trails of length k starting and ending at fixed distinct vertices in the complete undirected graph on n labeled vertices, for n greater than or equal to 1 and k equal to 0 .. n*(n-1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-16.
  196. ^ Sloane, N. J. A. (ed.). "Sequence A000014 (Number of series-reduced trees with n nodes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  197. ^ Sloane, N. J. A. (ed.). "Sequence A054502 (Counting sequence for classification of nonattacking queens on n X n toroidal board.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
  198. ^ Sloane, N. J. A. (ed.). "Sequence A054500 (Indicator sequence for classification of nonattacking queens on n X n toroidal board.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
  199. ^ Sloane, N. J. A. (ed.). "Sequence A054501 (Multiplicity sequence for classification of nonattacking queens on n X n toroidal board.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
  200. ^ I. Rivin, I. Vardi and P. Zimmermann (1994). The n-queens problem. American Mathematical Monthly. Washington, D.C.: Mathematical Association of America. 101 (7): 629–639. doi:10.1080/00029890.1994.11997004 JSTOR 2974691
  201. ^ Sloane, N. J. A. (ed.). "Sequence A135388 (Number of (directed) Eulerian circuits on the complete graph K_{2n+1}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-16.
  202. ^ Sloane, N. J. A. (ed.). "Sequence A007082 (Number of Eulerian circuits on the complete graph K_{2n+1}, divided by (n-1)!^(2n+1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-16.
  203. ^ Sloane, N. J. A. (ed.). "Sequence A357887 (Triangle read by rows: T(n,k) is the number of circuits of length k in the complete undirected graph on n labeled vertices, for n greater than or equal to 1 and k equal to 0 .. n(n-1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-16.
  204. ^ Sloane, N. J. A. (ed.). "Sequence A327070 (Number of non-connected simple labeled graphs covering n vertices.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-28.
  205. ^ Sloane, N. J. A. (ed.). "Sequence A141580 (Number of unlabeled non-mating graphs with n vertices.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  206. ^ Sloane, N. J. A. (ed.). "Sequence A290056 (Number of cliques in the n-triangular graph.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  207. ^ Sloane, N. J. A. (ed.). "Sequence A000568 (Number of outcomes of unlabeled n-team round-robin tournaments)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-15.
  208. ^ Royle, Gordon F.; Praeger, Cheryl E.; Glasby, S.P.; Freedman, Saul D.; Devillers, Alice (2023). "Tournaments and Graphs are Equinumerous". Journal of Algebraic Combinatorics. 57 (2). Heidelberg: Springer: 515–524. arXiv:2204.01947. doi:10.1007/s10801-022-01197-0. MR 4570411. S2CID 247958071. Zbl 1512.05199.
  209. ^ Sloane, N. J. A. (ed.). "Sequence A002190 (Sum_{n greater than or equal to 0} a(n)*x^n/n!^2 equal to -log(BesselJ(0,2*sqrt(x))).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  210. ^ Aval, Jean-Christophe; Boussicault, Adrien; Delcroix-Oger, Bérénice; et al. (2021). "Non-ambiguous trees: new results and generalization". European Journal of Combinatorics. 95. Amsterdam: Elsevier: 1–28. arXiv:1511.09455. doi:10.1016/j.ejc.2021.103331. MR 4111347. S2CID 265038614. Zbl 1440.05061.
  211. ^ Aval, Jean-Christophe; Boussicault, Adrien; Bouvel, Mathilde; Silimbani, Matteo (2014). "Combinatorics of non-ambiguous trees". Advances in Applied Mathematics. 56. Amsterdam: Elsevier: 78–108. arXiv:1305.3716. doi:10.1016/j.aam.2013.11.004. MR 3194208. S2CID 14989102. Zbl 1300.05127.
  212. ^ Sloane, N. J. A. (ed.). "Sequence A186725 (Number of connected regular graphs with n nodes and girth at least 5.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  213. ^ Sloane, N. J. A. (ed.). "Sequence A093134 (A Jacobsthal trisection.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  214. ^ Sloane, N. J. A. (ed.). "Sequence A367707 (Number of degree 6 vertices in the n-Menger sponge graph.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  215. ^ Sloane, N. J. A. (ed.). "Sequence A078469 (Number of different compositions of the ladder graph L_n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  216. ^ Sloane, N. J. A. (ed.). "Sequence A137883 (Number of (directed) Hamiltonian paths in the n-Möbius ladder graph.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  217. ^ Sloane, N. J. A. (ed.). "Sequence A124350 (a(n) equal to 4*n*(floor(n^2/2)+1). For n greater than or equal to 3, this is the number of directed Hamiltonian paths on the n-prism graph.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  218. ^ Sloane, N. J. A. (ed.). "Sequence A160172 (T-toothpick sequence (see Comments lines for definition).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  219. ^ Sloane, N. J. A. (ed.). "Sequence A118746 (Decimal expansion of onset of logistic map 7-bifurcation.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-29.
  220. ^ a b Sloane, N. J. A. (ed.). "Sequence A054377 (Primary pseudoperfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
  221. ^ Sloane, N. J. A. (ed.). "Sequence A007304 (Sphenic numbers: products of 3 distinct primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
  222. ^ Kellner, Bernd C. (2005). "The equation denom(Bn) = n has only one solution" (PDF). Bernouilli (Preprint). Göttingen, GER. pp. 1, 2. S2CID 124553392.
  223. ^ Sloane, N. J. A. (ed.). "Sequence A027642 (Denominator of Bernoulli number B_n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-12.
  224. ^ Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers: numbers k such that usigma(k) - k is k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-12.
  225. ^ Moree, Pieter (2004). "Convoluted Convolved Fibonacci Numbers" (PDF). Journal of Integer Sequences. 7 (2). Waterloo, Ont., CA: University of Waterloo David R. Cheriton School of Computer Science: 13 (Article 04.2.2). arXiv:math.CO/0311205. Bibcode:2004JIntS...7...22M. MR 2084694. S2CID 14126332. Zbl 1069.11004.
  226. ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A001629 (Self-convolution of Fibonacci numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  227. ^ Belbachir, Hacène; Djellal, Toufik; Luque, Jean-Gabriel (2023). "On the self-convolution of generalized Fibonacci numbers". Quaestiones Mathematicae. 46 (5). Oxfordshire, UK: Taylor & Francis: 841–854. arXiv:1703.00323. doi:10.2989/16073606.2022.2043949. MR 4592901. S2CID 119150217. Zbl 07707543.((cite journal)): CS1 maint: Zbl (link)
  228. ^ Berndt, Bruce C.; Chan, Heng Huat (1999). "Ramanujan and the modular j-invariant". Canadian Mathematical Bulletin. 42 (4): 427–440. doi:10.4153/CMB-1999-050-1. MR 1727340. S2CID 1816362.
  229. ^ Sloane, N. J. A. (ed.). "Sequence A004009 (Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
  230. ^ Sloane, N. J. A. (ed.). "Sequence A013973 (Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
  231. ^ Sloane, N. J. A. (ed.). "Sequence A006352 (Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
  232. ^ Hartshorne, Robin (1977). "Chapter 4: Curves". Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. Berlin: Springer-Verlag. pp. 316–321. doi:10.1007/978-1-4757-3849-0. ISBN 978-1-4419-2807-8. MR 1567149. OCLC 13348052. S2CID 5695770. Zbl 0367.14001.
  233. ^ Sloane, N. J. A. (ed.). "Sequence A060295 (Decimal expansion of exp(Pi*sqrt(163)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-18.
  234. ^ Barrow, John D. (2002). "The Constants of Nature". The Fundamental Constants. London: Jonathan Cape. p. 72. doi:10.1142/9789812818201_0001. ISBN 0-224-06135-6. S2CID 125272999.
  235. ^ a b c d e f g Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. 1–255. ISBN 978-0-19-280722-9. MR 2215662. OCLC 180766312. Zbl 1113.00002.
  236. ^ Ronan, Mark (6 March 2013). "163 and the Monster". Mark Ronan. Retrieved 2023-08-17.
  237. ^ a b c d Jansen, Christoph (2005). "The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups". LMS Journal of Computation and Mathematics. 8. London: London Mathematical Society: 122−123. doi:10.1112/S1461157000000930. MR 2153793. S2CID 121362819. Zbl 1089.20006.
  238. ^ Griess, Jr., Robert L. (1998). Twelve Sporadic Groups. Springer Monographs in Mathematics. Berlin: Springer-Verlag. pp. 54–79. ISBN 9783540627784. MR 1707296. OCLC 38910263. Zbl 0908.20007.
  239. ^ a b Madore, David Alexander (22 January 2003). "Orders of non abelian simple groups". www.madore.org. Retrieved 2024-01-10.
  240. ^ a b Sloane, N. J. A. (ed.). "Sequence A109379 (Orders of non-cyclic simple groups (with repetition).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-10.
  241. ^ Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-09.
  242. ^ a b Sloane, N. J. A. (ed.). "Sequence A067698 (Positive integers such that sigma(n) greater than or equal to exp(gamma) * n * log(log(n)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-09.
  243. ^ a b Sloane, N. J. A. (ed.). "Sequence A048242 (Numbers that are not the sum of two abundant numbers (not necessarily distinct).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-09.
  244. ^ Dietrich, Heiko; Lee, Melissa; Popiel, Tomasz (6 December 2023). "The maximal subgroups of the Monster". pp. 1–23. arXiv:2304.14646 [GR math. GR]. Bibcode:2023arXiv230414646D.
  245. ^ Griess, Jr., Robert L. (1982). "The Friendly Giant". Inventiones Mathematicae. 69: 91−96. Bibcode:1982InMat..69....1G. doi:10.1007/BF01389186. hdl:2027.42/46608. MR 0671653. S2CID 123597150. Zbl 0498.20013.
  246. ^ Ogg, A. P. (1981). "Modular Functions". In Cooperstein, Bruce; Mason, Geoffrey (eds.). The Santa Cruz Conference on Finite Groups. Proceedings of Symposia in Pure Mathematics. Vol. 37. Providence, RI: American Mathematical Society. pp. 521–532. doi:10.1090/PSPUM/037. ISBN 0-8218-1440-0. MR 0604631. Zbl 0443.00007.
  247. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes: primes dividing order of Monster simple group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-26.
  248. ^ Sloane, N. J. A. (ed.). "Sequence A216071 (Brocard's problem: positive integers m such that m^2 equal to n! + 1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.
  249. ^ Sloane, N. J. A. (ed.). "Sequence A085692 (Brocard's problem: squares which can be written as n!+1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.
  250. ^ Sloane, N. J. A. (ed.). "Sequence A004394 (Superabundant [or super-abundant] numbers: n such that sigma(n)/n greater than sigma(m)/m for all m less than n, sigma(n) being A000203(n), the sum of the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.
  251. ^ Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.
  252. ^ a b Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011). "Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis". Integers. 11 (6). Berlin: De Gruyter: 755 (A33). arXiv:1110.5078. doi:10.1515/INTEG.2011.057. MR 2798609. S2CID 14573608. Zbl 1235.11082.
  253. ^ Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-16.
  254. ^ Sloane, N. J. A. (ed.). "Sequence A002110 (Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-16.
  255. ^ Sloane, N. J. A. (ed.). "Sequence A000332 (Binomial coefficient binomial(n,4) equal to n*(n-1)*(n-2)*(n-3)/24.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-16.
  256. ^ Deshouillers, Jean-Marc; Granville, Andrew; Narkiewicz, Władysław; Pomerance, Carl (1993). "An upper bound in Goldbach's problem". Mathematics of Computation. 61 (203). Washington, D.C.: American Mathematical Society: 209–213. Bibcode:1993MaCom..61..209D. doi:10.1090/S0025-5718-1993-1202609-9. MR 1202609. S2CID 51831845. Zbl 0780.11047.
  257. ^ Sloane, N. J. A. (ed.). "Sequence A050381 (Number of series-reduced planted trees with n leaves of 2 colors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  258. ^ Sloane, N. J. A. (ed.). "Sequence A074127 (Group the composite numbers so that the sum of the n-th group is a multiple of the n-th prime: (4), (6), (8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22), (24, 25), (26, 27, 28, 30, 32), (33, 34, ...), ... Sequence gives the group sum divided by n-th prime for the n-th group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  259. ^ Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime (Lesser of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-06.
  260. ^ Sloane, N. J. A. (ed.). "Sequence A002410 (Nearest integer to imaginary part of n-th zero of Riemann zeta function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-16.
  261. ^ Sloane, N. J. A. (ed.). "Sequence A013629 (Floor of imaginary parts of nontrivial zeros of Riemann zeta function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-07.
  262. ^ Odlyzko, Andrew. "The first 100 (non trivial) zeros of the Riemann Zeta function [AT&T Labs]". Andrew Odlyzko: Home Page. UMN CSE. Retrieved 2024-01-16.
  263. ^ a b Klaise, Janis (2012). Orders in Quadratic Imaginary Fields of small Class Number (PDF) (MMath thesis). University of Warwick Centre for Complexity Science. pp. 1–24. S2CID 126035072.
  264. ^ Sloane, N. J. A. (ed.). "Sequence A258706 (Absolute primes: every permutation of digits is a prime. Only the smallest representative of each permutation class is shown.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-13.
  265. ^ Sloane, N. J. A. (ed.). "Sequence A059408 (a(n+1) as the a(n)-th composite and a(1) equal to 13.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-19.
  266. ^ Weisstein, Eric W. "Almost integer". MathWorld -- A WolframAlpha Resource. Retrieved 2023-10-18.
    The list below gives numbers of the form x ≡ eπ√n for n ≤ 1000 for which |nint(x) − x| ≤ 10−3.
    n |nint(x) − x|
    25 −0.00066
    37 −0.000022
    43 −0.00022
    58 −1.8×10−7
    67 −1.3×10−6
    74 −0.00083
    148 0.00097
    163 −7.5×10−13
    232 −7.8×10−6
    268 0.00029
    522 −0.00015
    652 1.6×10−10
    719 −0.000013
  267. ^ a b Sloane, N. J. A. (ed.). "Sequence A014603 (Discriminants of imaginary quadratic fields with class number 2 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-18.
  268. ^ Sloane, N. J. A. (ed.). "Sequence A032354 (j-invariants for orders of class number 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-26.
  269. ^ Zagier, Don (2008). "Elliptic Modular Forms and Their Applications". In Ranestad, Kristian (ed.). The 1-2-3 of Modular Forms. Universitext (1st ed.). Berlin: Springer-Verlag. pp. 70−71. doi:10.1007/978-3-540-74119-0_1. ISBN 978-3-540-74117-6. MR 2409678. OCLC 173239471. Zbl 1259.11042.
  270. ^ Arpin, Sarah; Camacho-Navarro, Catalina; et al. (2021). "Adventures in Supersingularland". Experimental Mathematics. 32 (2). Oxfordshire, UK: Taylor & Francis: 245−256. arXiv:1909.07779. Bibcode:2019arXiv190907779A. doi:10.1080/10586458.2021.1926009. MR 4592945. S2CID 202583238. Zbl 1517.94057.
  271. ^ a b Weisstein, Eric W. (2003). Weisstein, Eric W. (ed.). CRC Concise Encyclopedia of Mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC. pp. 57–59. doi:10.1201/9781420035223. ISBN 9781584883470. JSTOR 20453513. MR 1944431. OCLC 50252094. S2CID 116679721. Zbl 1079.00009.
  272. ^ Sloane, N. J. A. (ed.). "Sequence A329191 (The prime divisors of the orders of the sporadic finite simple groups.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-09.
  273. ^ Sloane, N. J. A. (ed.). "Sequence A113907 (Dimensions of the five sporadic Lie groups.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-09.
  274. ^ Sloane, N. J. A. (ed.). "Sequence A048050 (Chowla's function: sum of divisors of n except for 1 and n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-09.
  275. ^ Sloane, N. J. A. (ed.). "Sequence A006203 (Discriminants of imaginary quadratic fields with class number 3 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-23.
  276. ^ Sloane, N. J. A. (ed.). "Sequence A127106 (Numbers n such that n^2 divides 6^n-1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-14.
  277. ^ Sloane, N. J. A. (ed.). "Sequence A013658 (Discriminants of imaginary quadratic fields with class number 4 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-14.
  278. ^ Watkins, Mark (2004). "Class numbers of imaginary quadratic fields". Mathematics of Computation. 73 (246). Providence, RI: American Mathematical Society: 907–938. doi:10.1090/S0025-5718-03-01517-5. JSTOR 4099810. MR 2031415. Zbl 1050.11095.
  279. ^ Sloane, N. J. A. (ed.). "Sequence A038552 (Largest squarefree number k such that Q(sqrt(-k)) has class number n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-23.
  280. ^ a b Sloane, N. J. A. (ed.). "Sequence A004011 (Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-23.
  281. ^ a b Chun, Ji Hoon (2019). "Sphere Packings [2013]" (PDF). In Tsfasman, Michael; et al. (eds.). Algebraic Geometry Codes: Advanced Chapters. Mathematical Surveys and Monographs. Vol. 238. Providence, RI: American Mathematical Society. pp. 229−278. doi:10.1090/surv/238. ISBN 978-1-4704-5263-6. MR 3966406. S2CID 182109921. Zbl 1422.14004.
  282. ^ a b c Sloane, N. J. A. (ed.). "Sequence A004013 (Theta series of body-centered cubic (b.c.c.) lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-28.
  283. ^ a b Sloane, N. J. A. (ed.). "Sequence A004014 (Norms of vectors in the b.c.c. lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-28.
  284. ^ a b Sloane, N. J. A. (ed.). "Sequence A000118 (Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-23.
  285. ^ a b c d Coxeter, H. S. M. (1948). Regular Polytopes (1st ed.). London: Methuen & Co. pp. 1–321. ISBN 9780521201254. MR 0027148. OCLC 472190910.
  286. ^ Pekonen, Osmo (2018). "Which Integer Is the Most Mysterious?". In Sarhangi, Reza (ed.). Bridges: Mathematics, Art, Music, Architecture, Education, Culture. Proceedings of the Bridges Conference. p. 441. ISBN 9781938664274. ISSN 1590-5896. OCLC 7788577569. S2CID 133028298.
  287. ^ a b He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional". pp. 1–49. arXiv:1505.06742 [math.AG]. S2CID 117030188.
    "Of particular curiosity is the less well-known fact – in parallel to the above identity – that the constant term of the j-invariant, viz., 744, satisfies 744 = 3 × 248. The number 248 is, of course, the dimension of the adjoint of the largest exceptional algebra [𝖊8]. In fact, that j should encode the presentations of [𝖊8] was settled long before the final proof of the Moonshine conjectures. This relationship between the largest sporadic group and the largest exceptional algebra would connect the McKay Correspondence to Moonshine and thereby weave another beautiful thread into the panoramic tapestry of mathematics.": p.4 
    "16 Incidentally, the reader is also alerted to the curiosity that σ1(240) = 744.": p.21 (note) 
  288. ^ Sloane, N. J. A. (ed.). "Sequence A007245 (McKay-Thompson series of class 3C for the Monster group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
    j(𝜏)1/3 = q −1/3 (1 + 248q + 4124q2 + 34752q3 + ...)
  289. ^ a b Gannon, Terry (2006). "Introduction: glimpses of the theory beneath Monstrous Moonshine" (PDF). Moonshine beyond the monster: The bridge connecting algebra, modular forms and physics. Cambridge Monographs on Mathematical Physics. Cambridge, MA: Cambridge University Press. pp. 1–15. ISBN 978-0-521-83531-2. MR 2257727. OCLC 1374925688. Zbl 1146.11026.
    "In particular, 4124 = 3875 + 248 + 1 and 34752 = 30380 + 3875 + 2 · 248 + 1, where 248, 3875 and 30380 are all dimensions of irreducible representations of E8().": p.6 
  290. ^ Sloane, N. J. A. (ed.). "Sequence A121732 (Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.
  291. ^ Gaberdiel, Matthias R. (2007). "Constraints on extremal self-dual CFTs". Journal of High Energy Physics. 2007 (11, 087). Springer: 10–11. arXiv:0707.4073. Bibcode:2007JHEP...11..087G. doi:10.1088/1126-6708/2007/11/087. MR 2362062. S2CID 16635058.
  292. ^ Conway, John H.; Sloane, N. J. A. (1988). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. New York, NY: Springer. doi:10.1007/978-1-4757-2016-7. eISSN 2196-9701. ISBN 978-1-4757-2016-7. MR 1541550.
  293. ^ Coxeter, H. S. M. (1998). "Seven Cubes and Ten 24-Cells" (PDF). Discrete & Computational Geometry. 19 (2): 156–157. doi:10.1007/PL00009338. S2CID 206861928. Zbl 0898.52004.
  294. ^ Coxeter, H. S. M.; Shephard, G.C. (1992). "Portraits of a Family of Complex Polytopes". Leonardo. 25 (3/4): 243–244. doi:10.2307/1575843. JSTOR 1575843. S2CID 124245340. Zbl 0803.51023.
  295. ^ Baez, John C. (2018). "From the Icosahedron to E8". London Mathematical Society Newsletter. 476: 18–23. arXiv:1712.06436. MR 3792329. S2CID 119151549. Zbl 1476.51020.
  296. ^ Hart, George W. (1998). "Icosahedral Constructions" (PDF). In Sarhangi, Reza (ed.). Bridges: Mathematical Connections in Art, Music, and Science. Proceedings of the Bridges Conference. Winfield, Kansas. p. 196. ISBN 978-0966520101. OCLC 59580549. S2CID 202679388.((cite book)): CS1 maint: location missing publisher (link)
  297. ^ Wilson, Robert A. (2009). "Octonions and the Leech lattice". Journal of Algebra. 322 (6): 2186–2190. doi:10.1016/j.jalgebra.2009.03.021. MR 2542837.
  298. ^ Schellekens, Adrian Norbert (1993). "Meromorphic c = 24 conformal field theories". Communications in Mathematical Physics. 153 (1). Berlin: Springer: 159–185. arXiv:hep-th/9205072. Bibcode:1993CMaPh.153..159S. doi:10.1007/BF02099044. MR 1213740. S2CID 250425623. Zbl 0782.17014.
  299. ^ Möller, Sven; Scheithauer, Nils R. (2023). "Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra". Annals of Mathematics. 197 (1). Princeton University & the Institute for Advanced Study: 261–285. arXiv:1910.04947. Bibcode:2019arXiv191004947M. doi:10.4007/annals.2023.197.1.4. MR 4513145. S2CID 204401905. Zbl 07623024.((cite journal)): CS1 maint: Zbl (link)
  300. ^ Griess, Jr., Robert L.; Lam, Ching Hung (2011). "A moonshine path from E8 to the Monster" (PDF). Journal of Pure and Applied Algebra. 215 (5): 930–931. doi:10.1016/j.jpaa.2010.07.001. MR 2747229. S2CID 123613651. Zbl 1213.17028.
  301. ^ Griess Jr., Robert L.; Lam, Ching Hung (2011). "A moonshine path for 5 A and associated lattices of ranks 8 and 16". Journal of Algebra. 331. Amsterdam: Elsevier: 348. arXiv:1006.3907. Bibcode:2010arXiv1006.3907G. doi:10.1016/j.jalgebra.2010.11.013. MR 2774662. S2CID 119679255. Zbl 1283.20009.
  302. ^ Harada, Masaaki; Lam, Ching Hung; Munemasa, Akihiro (2013). "Residue codes of extremal Type II Z4-codes and the moonshine vertex operator algebra". Mathematische Zeitschrift. 274 (1). Springer: 691, 698–699. arXiv:1005.1144. Bibcode:2010arXiv1005.1144H. doi:10.1007/s00209-012-1091-z. MR 3054350. S2CID 253730965. Zbl 1283.94112.
  303. ^ Pless, V.; Sloane, N. J. A. (1975). "On the classification and enumeration of self-dual codes". Journal of Combinatorial Theory. Series A. 18 (3). Amsterdam: Elsevier: 313–335. doi:10.1016/0097-3165(75)90042-4. MR 0376232. S2CID 16303075. Zbl 0305.94011.
  304. ^ Van Ekeren, Jethro; Lam, Ching Hung; Möller, Sven; Shimakura, Hiroki (2021). "Schellekens' list and the very strange formula". Advances in Mathematics. 380. Amsterdam: Elsevier: 1–34 (107567). arXiv:2005.12248. doi:10.1016/j.aim.2021.107567. MR 4200469. S2CID 218870375. Zbl 1492.17027.
  305. ^ Betsumiya, Koichi; Lam, Ching Hung; Shimakura, Hiroki (2023). "Automorphism Groups and Uniqueness of Holomorphic Vertex Operator Algebras of Central Charge 24". Communications in Mathematical Physics. 399 (3). Berlin: Springer: 1773–1810. arXiv:2203.15992. Bibcode:2023CMaPh.399.1773B. doi:10.1007/s00220-022-04585-6. MR 4580534. S2CID 247793117. Zbl 1523.17056.
  306. ^ Sloane, N. J. A. (ed.). "Sequence A129863 (Sums of three consecutive pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  307. ^ Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  308. ^ Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers: a(n) equal to n^2*(n+1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-09.
  309. ^ Taneja, Inder J. (2016). "Selfie Numbers - IV: Addition, Subtraction and Factorial" (PDF). RGMIA Research Report Collection. 19: 1−4 (Article 163).
  310. ^ Sloane, N. J. A. (ed.). "Sequence A193069 (Pretty wild narcissistic numbers - numbers that pwn: - an integer n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ ! and the radical symbol. Concatenation is allowed.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-12.
  311. ^ Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  312. ^ Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
  313. ^ Sloane, N. J. A. (ed.). "Sequence A122875 (Numbers whose square is undulating.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
  314. ^ Sloane, N. J. A. (ed.). "Sequence A016073 (Undulating squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
  315. ^ Pickover, Clifford A. (1995). "Chapter 20: The Undulation of the Monks". Keys to Infinity. New York, NY: John Wiley & Sons, Inc. pp. xviii, 1−332. ISBN 978-0471193340. OCLC 31710129. S2CID 121761396.
  316. ^ a b Sloane, N. J. A. (ed.). "Sequence A164843 (The smallest magic constant of an n X n magic square with distinct prime entries.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-05.
  317. ^ Makarova, Natalia V. "Least Magic Squares of Prime Numbers (Part I)". klassikpoez.narod.ru (in Russian). Archived from the original on 2023-12-08. Archived translation in English. Retrieved 2023-12-07.
  318. ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
  319. ^ Sloane, N. J. A. (ed.). "Sequence A187781 (Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-23.
  320. ^ Sloane, N. J. A. (ed.). "Sequence A279019 (Least possible number of diagonals of simple convex polyhedron with n faces.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-10.
  321. ^ Sloane, N. J. A. (ed.). "Sequence A002839 (Number of simple perfect squared rectangles of order n up to symmetry.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  322. ^ Sloane, N. J. A. (ed.). "Sequence A326118 (a(n) is the largest number of squares of unit area connected only at corners and without holes that can be inscribed in an n X n square.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-20.
  323. ^ Sloane, N. J. A. (ed.). "Sequence A272390 (Longest side of primitive Heronian tetrahedron with 4 congruent triangle faces.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-09.
  324. ^ Sloane, N. J. A. (ed.). "Sequence A272388 (Longest side of Heronian tetrahedron.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-09.


Categories