61 (number)

Summary

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

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

In mathematics edit

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

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

61 is also the largest prime factor in Descartes number,[17]

 

This number would be the only known odd perfect number if one of its composite factors (22021 = 192 × 61) were prime.[18]

61 is the largest prime number (less than the largest supersingualar prime, 71) that does not divide the order of any sporadic group (including any of the pariahs).

The exotic sphere   is the last odd-dimensional sphere to contain a unique smooth structure;  ,   and   are the only other such spheres.[19][20]

In science edit

Astronomy edit

In other fields edit

Sixty-one is:

In sports edit

Notelist edit

  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 edit

  1. ^ Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers: a(n) is 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z equal to Y+1) ordered by increasing Z; then sequence gives Z values.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-09.
  2. ^ "Sloane's A062786 : Centered 10-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  3. ^ "Sloane's A003215 : Hex (or centered hexagonal) 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. ^ Holdener, Judy; Rachfal, Emily (2019). "Perfect and Deficient Perfect Numbers". The American Mathematical Monthly. Mathematical Association of America. 126 (6): 541–546. doi:10.1080/00029890.2019.1584515. MR 3956311. S2CID 191161070. Zbl 1477.11012 – via Taylor & Francis.
  18. ^ Sloane, N. J. A. (ed.). "Sequence A222262 (Divisors of Descarte's 198585576189.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
  19. ^ 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.
  20. ^ 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.
  21. ^ Hoyle, Edmund Hoyle's Official Rules of Card Games pub. Gary Allen Pty Ltd, (2004) p. 470
  22. ^ MySQL Reference Manual – JOIN clause
  • R. Crandall and C. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer, NY, 2005, p. 79.

External links edit