61 (number)

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

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

,[4] and the fourth 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.

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,

[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.

Sixty-one is the exponent of the ninth Mersenne prime,

[9] and the next candidate exponent for a potential fifth double Mersenne prime:

[10] 61 is also the largest prime factor in Descartes number,[11]

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

[12] 61 is the largest prime number (less than the largest supersingular prime, 71) that does not divide the order of any sporadic group (including any of the pariahs).

is the last odd-dimensional sphere to contain a unique smooth structure;