A113957 Sum of the divisors of n which are not divisible by 7.

1, 3, 4, 7, 6, 12, 1, 15, 13, 18, 12, 28, 14, 3, 24, 31, 18, 39, 20, 42, 4, 36, 24, 60, 31, 42, 40, 7, 30, 72, 32, 63, 48, 54, 6, 91, 38, 60, 56, 90, 42, 12, 44, 84, 78, 72, 48, 124, 1, 93, 72, 98, 54, 120, 72, 15, 80, 90, 60, 168, 62, 96, 13, 127, 84, 144, 68, 126, 96, 18, 72

Offset: 1

Author

Michael Somos, Nov 10 2005

Keywords

• nonn
• mult

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 467, Entry 5(i).

Links

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

Crossrefs

Cf. A028594(n)=4*a(n) if n>0.

Cf. A244600.

Programs

Mathematica

f[p_, e_] := If[p == 7, 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, sigma(n/7^valuation(n,7)))

Formula

a(n) is multiplicative and a(p^e) = 1, if p=7, a(p^e) = (p^(e+1)-1)/(p-1) otherwise.

G.f.: ((theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z))^2-1)/4.

L.g.f.: log(Product_{k>=1} (1 - x^(7*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018

Sum_{k=1..n} a(k) ~ (Pi^2/14) * n^2. - Amiram Eldar, Oct 04 2022

Dirichlet g.f. (1-7^(1-s))*zeta(s)*zeta(s-1). - R. J. Mathar, May 17 2023