A325469 a(n) is the number of divisors d of n such that d divides sigma(d).

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Jaroslav Krizek, Aug 16 2019

Keywords

• nonn

Comments

Sequence of the smallest numbers m with n divisors d such that d divides sigma(d) for n >= 1: 1, 6, 84, 672, 3360, 30240, 393120, ...

Examples

			For n = 12, divisors d of 12: 1, 2, 3, 4, 6, 12; corresponding sigma(d): 1, 3, 4, 7, 12, 28; d divides sigma(d) for 2 divisors d: 1 and 6; a(12) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A000203, A097603, A325470, A325471, A335830.

Programs

Magma

[#[d: d in Divisors(n) | IsIntegral(SumOfDivisors(d) / d)] : n in [1..100]];

Mathematica

a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], #] &];  Array[a, 100] (* Amiram Eldar, Aug 17 2019 *)

PARI

a(n)={sumdiv(n, d, sigma(d) % d == 0)} \\ Andrew Howroyd, Aug 16 2019

Formula

a(A097603(n)) > 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A335830. - Amiram Eldar, Apr 16 2025

Extensions

More terms from Antti Karttunen, Aug 22 2019