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

%I A285896 #31 Dec 30 2022 03:55:38
%S A285896 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,
%T A285896 42,40,56,30,60,32,63,48,54,40,91,38,60,56,75,42,96,44,84,65,72,48,
%U A285896 124,57,75,72,98,54,120,60,120,80,90,60,140,62,96,104,127,70,144
%N A285896 Sum of divisors d of n such that n/d is not congruent to 0 mod 5.
%H A285896 Seiichi Manyama, <a href="/A285896/b285896.txt">Table of n, a(n) for n = 1..10000</a>
%F A285896 a(n) = (A000203(5*n)-A000203(n))/5.
%F A285896 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
%F A285896 From _Amiram Eldar_, Oct 30 2022: (Start)
%F A285896 Multiplicative with a(5^e) = 5^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 5.
%F A285896 Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/25 = 0.789568... . (End)
%F A285896 Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/5^s). - _Amiram Eldar_, Dec 30 2022
%e A285896 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.
%t A285896 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 *)
%o A285896 (PARI) a(n)=sumdiv(n, d, if(n/d%5, d, 0)); \\ _Andrew Howroyd_, Jul 20 2018
%Y A285896 Cf. A002131 (k=2), A078708 (k=3), A285895 (k=4), this sequence (k=5).
%Y A285896 Cf. A000203.
%K A285896 nonn,mult
%O A285896 1,2
%A A285896 _Seiichi Manyama_, Apr 28 2017
%E A285896 Keyword:mult added by _Andrew Howroyd_, Jul 20 2018