A286758 Numbers k such that sigma(k) divides sigma(k!).

1, 2, 3, 5, 7, 8, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Author

Jaroslav Krizek, May 14 2017

Keywords

• nonn

Comments

Conjecture: If p is Fermat prime > 3 from A019434 both values sigma((p-1)!) mod sigma(p-1) and sigma(T(p-1)) mod sigma(p-1) are not 0, where T(n) is the n-th triangular number A000217(n) and n! is the factorial number A000142(n).

Examples

			8 is a term because sigma(8!) / sigma(8) = sigma(40320) / sigma(8) = 159120 / 15 = 10608 (integer).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000 (first 1000 terms from Jaroslav Krizek)

Crossrefs

Complement of A262812.

All primes (A000040) are terms.

Cf. A000142, A000203.

Programs

Magma

[n: n in [1..100] | (SumOfDivisors(Factorial(n))) mod SumOfDivisors(n) eq 0];

Mathematica

A286758Q[k_] := PrimeQ[k] || Divisible[DivisorSigma[1, k!], DivisorSigma[1, k]];
Select[Range[100], A286758Q] (* Paolo Xausa, Jul 31 2025 *)