Founded | 1964 |
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Predecessor(s) | Handbook of Integer Sequences, Encyclopedia of Integer Sequences |
Created by | Neil Sloane |
Chairman | Neil Sloane |
President | Russ Cox |
URL | oeis |
Commercial | No^{[1]} |
Registration | Optional^{[2]} |
Launched | 1996 |
Content license | Creative Commons CC BY-NC 3.0^{[3]} |
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009.^{[4]} Sloane is chairman of the OEIS Foundation.
OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. As of January 2022^{[ref]}, it contains over 350,000 sequences, making it the largest database of its kind.
Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields.
Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics.^{[5]} The database was at first stored on punched cards. He published selections from the database in book form twice:
These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service (August 1994), and soon after as a website (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.^{[6]} The database continues to grow at a rate of some 10,000 entries a year. Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database.^{[7]} In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000, which counts the marks on the Ishango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors.^{[8]} The 200,000th sequence, A200000, was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list,^{[9]}^{[10]} following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a special sequence for A200000.^{[11]} A300000 was defined in February 2018, and by end of July 2020 the database contained more than 336,000 sequences.
Besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of fractions are represented by two sequences (named with the keyword 'frac'): the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence, , is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 (A006842) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 (A006843). Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions (here 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, ... (A000796)), binary expansions (here 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, ... (A004601)), or continued fraction expansions (here 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, ... (A001203)).
The OEIS was limited to plain ASCII text until 2011, and it still uses a linear form of conventional mathematical notation (such as f(n) for functions, n for running variables, etc.). Greek letters are usually represented by their full names, e.g., mu for μ, phi for φ.
Every sequence is identified by the letter A followed by six digits, almost always referred to with leading zeros, e.g., A000315 rather than A315.
Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces.
In comments, formulas, etc., a(n)
represents the nth term of the sequence.
Zero is often used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n^{2} consecutive primes to form an n × n magic square of least magic constant, or 0 if no such magic square exists." The value of a(1) (a 1 × 1 magic square) is 2; a(3) is 1480028129. But there is no such 2 × 2 magic square, so a(2) is 0. This special usage has a solid mathematical basis in certain counting functions; for example, the totient valence function N_{φ}(m) (A014197) counts the solutions of φ(x) = m. There are 4 solutions for 4, but no solutions for 14, hence a(14) of A014197 is 0—there are no solutions.
Other values are also used, most commonly −1 (see A000230 or A094076).
The OEIS maintains the lexicographical order of the sequences, so each sequence has a predecessor and a successor (its "context").^{[12]} OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring all initial zeros and ones, and also the sign of each element. Sequences of weight distribution codes often omit periodically recurring zeros.
For example, consider: the prime numbers, the palindromic primes, the Fibonacci sequence, the lazy caterer's sequence, and the coefficients in the series expansion of . In OEIS lexicographic order, they are:
whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2.
Very early in the history of the OEIS, sequences defined in terms of the numbering of sequences in the OEIS itself were proposed. "I resisted adding these sequences for a long time, partly out of a desire to maintain the dignity of the database, and partly because A22 was only known to 11 terms!", Sloane reminisced.^{[13]} One of the earliest self-referential sequences Sloane accepted into the OEIS was A031135 (later A091967) "a(n) = n-th term of sequence A_{n} or –1 if A_{n} has fewer than n terms". This sequence spurred progress on finding more terms of A000022. A100544 lists the first term given in sequence A_{n}, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term a(1) of sequence A_{n} might seem a good alternative if it weren't for the fact that some sequences have offsets of 2 and greater. This line of thought leads to the question "Does sequence A_{n} contain the number n?" and the sequences A053873, "Numbers n such that OEIS sequence A_{n} contains n", and A053169, "n is in this sequence if and only if n is not in sequence A_{n}". Thus, the composite number 2808 is in A053873 because A002808 is the sequence of composite numbers, while the non-prime 40 is in A053169 because it's not in A000040, the prime numbers. Each n is a member of exactly one of these two sequences, and in principle it can be determined which sequence each n belongs to, with two exceptions (related to the two sequences themselves):
This entry, A046970, was chosen because it contains every field an OEIS entry can have.^{[14]}
A046970 Dirichlet inverse of the Jordan function J_2 (A007434).
1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576
OFFSET 1,2
COMMENTS B(n+2) = -B(n)*((n+2)*(n+1)/(4*Pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4*Pi^2)) * Sum_{j>=1} a(j)/j^(n+2).
Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002
REFERENCES M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.
LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
Wikipedia, Riemann zeta function.
FORMULA Multiplicative with a(p^e) = 1 - p^2.
a(n) = Sum_{d|n} mu(d)*d^2.
abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - Jon Perry, Aug 24 2010
From Wolfdieter Lang, Jun 16 2011: (Start)
Dirichlet g.f.: zeta(s)/zeta(s-2).
a(n) = J_{-2}(n)*n^2, with the Jordan function J_k(n), with J_k(1):=1. See the Apostol reference, p. 48. exercise 17. (End)
a(prime(n)) = -A084920(n). - R. J. Mathar, Aug 28 2011
G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017
EXAMPLE a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
a(4) = -3 because the divisors of 4 are {1, 2, 4} and mu(1)*1^2 + mu(2)*2^2 + mu(4)*4^2 = -3.
E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - Jon Perry, Aug 24 2010
G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + ...
MAPLE Jinvk := proc(n, k) local a, f, p ; a := 1 ; for f in ifactors(n)[2] do p := op(1, f) ; a := a*(1-p^k) ; end do: a ; end proc:
A046970 := proc(n) Jinvk(n, 2) ; end proc: # R. J. Mathar, Jul 04 2011
MATHEMATICA muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez)
Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(1 - x[[i]][[1]]^2)]; p}, {n, 1, 50, 1}]] (* Jon Perry, Aug 24 2010 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ d], {d, Divisors @ n}]] (* Michael Somos, Jan 11 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (1 - #[[1]]^2 & /@ FactorInteger @ n)] (* Michael Somos, Jan 11 2014 *)
PROG (PARI) A046970(n)=sumdiv(n, d, d^2*moebius(d)) \\ Benoit Cloitre
(Haskell)
a046970 = product . map ((1 -) . (^ 2)) . a027748_row
-- Reinhard Zumkeller, Jan 19 2012
(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - X*p^2) / (1 - X))[n])} /* Michael Somos, Jan 11 2014 */
CROSSREFS Cf. A007434, A027641, A027642, A063453, A023900.
Cf. A027748.
Sequence in context: A144457 A220138 A146975 * A322360 A058936 A280369
Adjacent sequences: A046967 A046968 A046969 * A046971 A046972 A046973
KEYWORD sign,easy,mult
AUTHOR Douglas Stoll, dougstoll(AT)email.msn.com
EXTENSIONS Corrected and extended by Vladeta Jovovic, Jul 25 2001
Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005
A059097 | Numbers n such that the binomial coefficient C(2n, n) is not divisible by the square of an odd prime. | Jan 1, 2001 |
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A060001 | Fibonacci(n)!. | Mar 14, 2001 |
A066288 | Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 24. | Jan 1, 2002 |
A075000 | Smallest number such that n · a(n) is a concatenation of n consecutive integers ... | Aug 31, 2002 |
A078470 | Continued fraction for ζ(3/2) | Jan 1, 2003 |
A080000 | Number of permutations satisfying −k ≤ p(i) − i ≤ r and p(i) − i | Feb 10, 2003 |
A090000 | Length of longest contiguous block of 1s in binary expansion of nth prime. | Nov 20, 2003 |
A091345 | Exponential convolution of A069321(n) with itself, where we set A069321(0) = 0. | Jan 1, 2004 |
A100000 | Marks from the 22000-year-old Ishango bone from the Congo. | Nov 7, 2004 |
A102231 | Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right. | Jan 1, 2005 |
A110030 | Number of consecutive integers starting with n needed to sum to a Niven number. | Jul 8, 2005 |
A112886 | Triangle-free positive integers. | Jan 12, 2006 |
A120007 | Möbius transform of sum of prime factors of n with multiplicity. | Jun 2, 2006 |
A016623 | 3, 8, 3, 9, 4, 5, 2, 3, 1, 2, ... | Decimal expansion of ln(93/2). |
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A046543 | 1, 1, 1, 3, 8, 3, 10, 1, 110, 3, 406, 3 | First numerator and then denominator of the central elements of the 1/3-Pascal triangle (by row). |
A035292 | 1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, ... | Number of similar sublattices of Z^{4} of index n^{2}. |
A046970 | 1, −3, −8, −3, −24, 24, −48, −3, −8, 72, ... | Generated from Riemann zeta function... |
A058936 | 0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840, 504, 420, 5760, 3360, 2688, 1260 |
Decomposition of Stirling's S(n, 2) based on associated numeric partitions. |
A002017 | 1, 1, 1, 0, −3, −8, −3, 56, 217, 64, −2951, −12672, ... | Expansion of exp(sin x). |
A086179 | 3, 8, 4, 1, 4, 9, 9, 0, 0, 7, 5, 4, 3, 5, 0, 7, 8 | Decimal expansion of upper bound for the r-values supporting stable period-3 orbits in the logistic map. |
In 2009, the OEIS database was used by Philippe Guglielmetti to measure the "importance" of each integer number.^{[19]} The result shown in the plot on the right shows a clear "gap" between two distinct point clouds,^{[20]} the "uninteresting numbers" (blue dots) and the "interesting" numbers that occur comparatively more often in sequences from the OEIS. It contains essentially prime numbers (red), numbers of the form a^{n} (green) and highly composite numbers (yellow). This phenomenon was studied by Nicolas Gauvrit, Jean-Paul Delahaye and Hector Zenil who explained the speed of the two clouds in terms of algorithmic complexity and the gap by social factors based on an artificial preference for sequences of primes, even numbers, geometric and Fibonacci-type sequences and so on.^{[21]} Sloane's gap was featured on a Numberphile video in 2013.^{[22]}