A173441 Number of divisors d of n such that sigma(d) divides n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Author

Jaroslav Krizek, Feb 18 2010

Keywords

• nonn

Comments

From Robert Israel, Oct 11 2017: (Start)

a(n) >= 1 since d=1 is always included.

a(n) = 1 if n is in A000961.

a(n) > 1 if n is in A097603. The first n not in A097603 such that a(n) > 1 is 117. (End)

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000961, A009242, A097603, A173442.

Programs

Maple

f:= proc(n) nops(select(t -> n mod numtheory:-sigma(t) = 0, numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Oct 11 2017

Mathematica

a[n_] := Select[Divisors[n], Divisible[n, DivisorSigma[1, #]]&] // Length;
Array[a, 100] (* Jean-François Alcover, Jun 05 2020 *)

PARI

a(n) = sumdiv(n, d, !(n % sigma(d))); \\ Michel Marcus, Oct 11 2017

Formula

a(n) = A000005(n) - A173442(n). - A-number inserted by R. J. Mathar, Mar 06 2010

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A009242(n) = 1.605582... . - Amiram Eldar, Mar 28 2024