← 60 61 62 →
Cardinalsixty-one
Ordinal61st
(sixty-first)
Factorizationprime
Prime18th
Divisors1, 61
Greek numeralΞΑ´
Roman numeralLXI
Binary1111012
Ternary20213
Senary1416
Octal758
Duodecimal5112
Hexadecimal3D16

61 (sixty-one) is the natural number following 60 and preceding 62.

In mathematics

61 is the 18th prime number, and a twin prime with 59. It is the sum of two consecutive squares, It is also a centered decagonal number,[1] a centered hexagonal number,[2] and a centered square number.[3]

61 is the fourth cuban prime of the form where ,[4] and the forth Pillai prime since is divisible by 61, but 61 is not one more than a multiple of 8.[5] It is also a Keith number, as it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61, ...[6]

61 is a unique prime in base 14, since no other prime has a 6-digit period in base 14, and palindromic in bases 6 (1416) and 60 (1160). It is the sixth up/down or Euler zigzag number.

61 is the smallest proper prime, a prime which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length where each digit (0, 1, ..., 9) appears in the repeating sequence the same number of times as does each other digit (namely, times).[7]: 166 

In the list of Fortunate numbers, 61 occurs thrice, since adding 61 to either the tenth, twelfth or seventeenth primorial gives a prime number[8] (namely 6,469,693,291; 7,420,738,134,871; and 1,922,760,350,154,212,639,131).

There are sixty-one 3-uniform tilings, where on the other hand, there are one hundred and fifty-one 4-uniform tilings[9] (with 61 the eighteenth prime number, and 151 the thirty-sixth, twice the index value).[10][a]

61 is the exponent of the ninth Mersenne prime, [15] and the next candidate exponent for a potential fifth double Mersenne prime: [16]

The exotic sphere is the last odd-dimensional sphere to contain a unique smooth structure; , and are the only other such spheres.[17][18]

In science

Astronomy

In other fields

See also: List of highways numbered 61

Sixty-one is:

In sports

Notelist

  1. ^ Otherwise, there are eleven total 1-uniform tilings (the regular and semiregular tilings), and twenty 2-uniform tilings (where 20 is the eleventh composite number;[11] together these values add to 31, the eleventh prime).[10][12] The sum of the first twenty integers is the fourth primorial 210,[13][14] equal to the product of the first four prime numbers, and 1, whose collective sum generated is 18.

References

  1. ^ "Sloane's A062786 : Centered 10-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  2. ^ "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  3. ^ "Sloane's A001844 : Centered square numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  4. ^ "Sloane's A002407 : Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  5. ^ "Sloane's A063980 : Pillai primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  6. ^ "Sloane's A007629 : Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  7. ^ Dickson, L. E., History of the Theory of Numbers, Volume 1, Chelsea Publishing Co., 1952.
  8. ^ "Sloane's A005235 : Fortunate numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A068599 (Number of n-uniform tilings.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-07.
  10. ^ a b Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-07.
  11. ^ 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 2024-01-07.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A299782 (a(n) is the total number of k-uniform tilings, for k equal to 1..n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-07.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers: a(n) is the binomial(n+1,2) equal to n*(n+1)/2 or 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-07.
  14. ^ 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 2024-01-07.
  15. ^ "Sloane's A000043 : Mersenne exponents". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  16. ^ "Mersenne Primes: History, Theorems and Lists". PrimePages. Retrieved 2023-10-22.
  17. ^ Wang, Guozhen; Xu, Zhouli (2017). "The triviality of the 61-stem in the stable homotopy groups of spheres". Annals of Mathematics. 186 (2): 501–580. arXiv:1601.02184. doi:10.4007/annals.2017.186.2.3. MR 3702672. S2CID 119147703.
  18. ^ Sloane, N. J. A. (ed.). "Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-22.
  19. ^ Hoyle, Edmund Hoyle's Official Rules of Card Games pub. Gary Allen Pty Ltd, (2004) p. 470
  20. ^ MySQL Reference Manual – JOIN clause