A285896 Sum of divisors d of n such that n/d is not congruent to 0 mod 5.

1, 3, 4, 7, 5, 12, 8, 15, 13, 15, 12, 28, 14, 24, 20, 31, 18, 39, 20, 35, 32, 36, 24, 60, 25, 42, 40, 56, 30, 60, 32, 63, 48, 54, 40, 91, 38, 60, 56, 75, 42, 96, 44, 84, 65, 72, 48, 124, 57, 75, 72, 98, 54, 120, 60, 120, 80, 90, 60, 140, 62, 96, 104, 127, 70, 144

Offset: 1

Author

Seiichi Manyama, Apr 28 2017

Keywords

• nonn
• mult

Examples

			The divisors of 10 are 1, 2, 5, and 10. 10/1 == 0 (mod 5) and 10/2 == 0 (mod 5). Hence, a(10) = 5 + 10 = 15.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002131 (k=2), A078708 (k=3), A285895 (k=4), this sequence (k=5).

Cf. A000203.

Programs

Mathematica

f[p_, e_] := If[p == 5, 5^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

PARI

a(n)=sumdiv(n, d, if(n/d%5, d, 0)); \\ Andrew Howroyd, Jul 20 2018

Formula

a(n) = (A000203(5*n)-A000203(n))/5.

G.f.: Sum_{k>=1} k*x^k*(1 + x^k + x^(2*k) + x^(3*k))/(1 - x^(5*k)). - Ilya Gutkovskiy, Sep 12 2019

From Amiram Eldar, Oct 30 2022: (Start)

Multiplicative with a(5^e) = 5^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 5.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/25 = 0.789568... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/5^s). - Amiram Eldar, Dec 30 2022

Extensions

Keyword:mult added by Andrew Howroyd, Jul 20 2018