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%I #42 May 17 2023 04:11:12
%S 1,2,2,3,1,4,2,4,3,2,2,6,2,4,2,5,2,6,2,3,4,4,2,8,1,4,4,6,2,4,2,6,4,4,
%T 2,9,2,4,4,4,2,8,2,6,3,4,2,10,3,2,4,6,2,8,2,8,4,4,2,6,2,4,6,7,2,8,2,6,
%U 4,4,2,12,2,4,2,6,4,8,2,5,5,4,2,12,2,4,4,8,2,6,4,6,4,4,2,12,2,6,6,3,2,8,2
%N Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = 25.
%C Number of divisors of n not congruent to 0 mod 5.
%H Amiram Eldar, <a href="/A035207/b035207.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(5^e)=1 and a(p^e)=e+1 for p<>5.
%F Moebius transform is period 5 sequence A011558. - _Michael Somos_, Oct 31 2006
%F G.f.: Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k))/(1 - x^(5*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F a(n) = tau(5*n) - tau(n). - _Ridouane Oudra_, Sep 05 2020
%F From _Amiram Eldar_, Nov 27 2022: (Start)
%F Dirichlet g.f.: zeta(s)^2 * (1 - 1/5^s).
%F Sum_{k=1..n} a(k) ~ (4*n*log(n) + (8*gamma + log(5) - 4)*n)/5, where gamma is Euler's constant (A001620). (End)
%p for n from 1 to 500 do a := ifactors(n):s := 1:for k from 1 to nops(a[2]) do p := a[2][k][1]:e := a[2][k][2]: if p=5 then b := 1:else b := e+1:fi:s := s*b:od:printf(`%d,`,s); od:
%t Table[Count[Divisors[n],_?(!Divisible[#,5]&)],{n,110}] (* _Harvey P. Dale_, Apr 08 2015 *)
%t f[5, e_] := 1; f[p_, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 26 2020 *)
%o (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, d%5>0))} /* _Michael Somos_, Oct 31 2006 */
%o (PARI) {a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/if(p==5, 1, 1-X))[n])} /* _Michael Somos_, Oct 31 2006 */
%o (Magma) [NumberOfDivisors(n)/Valuation(5*n, 5): n in [1..100]]; // _Vincenzo Librandi_, Jun 03 2019
%Y Cf. A000005 (tau), A001620, A035191, A069733.
%Y Cf. A116073 (sum of divisors of n not congruent to 0 mod 5).
%K nonn,mult,easy
%O 1,2
%A _N. J. A. Sloane_
%E Additional comments from _Vladeta Jovovic_, Oct 26 2001
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