A116073 Sum of the divisors of n that are not divisible by 5.

1, 3, 4, 7, 1, 12, 8, 15, 13, 3, 12, 28, 14, 24, 4, 31, 18, 39, 20, 7, 32, 36, 24, 60, 1, 42, 40, 56, 30, 12, 32, 63, 48, 54, 8, 91, 38, 60, 56, 15, 42, 96, 44, 84, 13, 72, 48, 124, 57, 3, 72, 98, 54, 120, 12, 120, 80, 90, 60, 28, 62, 96, 104, 127, 14, 144, 68, 126, 96, 24

Offset: 1

Author

Michael Somos, Feb 04 2006

Keywords

• nonn
• mult

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 463 Entry 4(i).

Links

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

Crossrefs

Cf. A028887(n) = 6*a(n) if n>0.

Cf. A145466.

Cf. A091703, A035207 (number of divisors of n that are not divisible by 5).

Programs

Mathematica

f[p_, e_] := If[p == 5, 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

PARI

a(n) = if(n<1, 0, sumdiv(n,d,(d%5>0)*d))

Formula

a(n) is multiplicative with a(5^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) otherwise.

G.f.: Sum_{k>0} k*x^k/(1-x^k) - 5*k*x^(5*k)/(1-x^(5*k)).

L.g.f.: log(Product_{k>=1} (1 - x^(5*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018

Sum_{k=1..n} a(k) ~ (Pi^2/15) * n^2. - Amiram Eldar, Oct 04 2022

Inverse Mobius transf. of A091703. Dirichlet g.f. (1-5^(1-s))*zeta(s-1)*zeta(s). - R. J. Mathar, May 17 2023