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

%I A113957 #20 May 17 2023 04:18:22
%S A113957 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,
%T A113957 42,40,7,30,72,32,63,48,54,6,91,38,60,56,90,42,12,44,84,78,72,48,124,
%U A113957 1,93,72,98,54,120,72,15,80,90,60,168,62,96,13,127,84,144,68,126,96,18,72
%N A113957 Sum of the divisors of n which are not divisible by 7.
%D A113957 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 467, Entry 5(i).
%H A113957 Seiichi Manyama, <a href="/A113957/b113957.txt">Table of n, a(n) for n = 1..10000</a>
%F A113957 a(n) is multiplicative and a(p^e) = 1, if p=7, a(p^e) = (p^(e+1)-1)/(p-1) otherwise.
%F A113957 G.f.: ((theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z))^2-1)/4.
%F A113957 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
%F A113957 Sum_{k=1..n} a(k) ~ (Pi^2/14) * n^2. - _Amiram Eldar_, Oct 04 2022
%F A113957 Dirichlet g.f. (1-7^(1-s))*zeta(s)*zeta(s-1). - _R. J. Mathar_, May 17 2023
%t A113957 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 *)
%o A113957 (PARI) a(n)=if(n<1, 0, sigma(n/7^valuation(n,7)))
%Y A113957 Cf. A028594(n)=4*a(n) if n>0.
%Y A113957 Cf. A244600.
%K A113957 nonn,mult
%O A113957 1,2
%A A113957 _Michael Somos_, Nov 10 2005