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A378008 a(n) = b(5*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 5; 5th column of A378007.

%I A378008 #12 Nov 14 2024 23:23:23
%S A378008 1,0,4,1,0,0,4,0,4,0,0,0,4,0,4,0,1,0,0,0,4,0,0,0,10,0,4,0,0,0,4,0,0,0,
%T A378008 0,4,4,0,4,0,0,0,4,0,0,0,0,0,4,0,4,1,0,0,4,0,4,0,0,0,0,0,4,0,0,0,4,0,
%U A378008 16,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,4,0,4,0,0,0,16
%N A378008 a(n) = b(5*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 5; 5th column of A378007.
%H A378008 Jianing Song, <a href="/A378008/b378008.txt">Table of n, a(n) for n = 0..10000</a>
%F A378008 a(n) = b(5*n+1), where {b(n)} is multiplicative with:
%F A378008 - b(5^e) = 0;
%F A378008 - for p == 1 (mod 5), b(p^e) = binomial(e+3,3) = (e+3)*(e+2)*(e+1)/6;
%F A378008 - for p == 4 (mod 5), b(p^e) = e/2 + 1 if e is even, and 0 otherwise;
%F A378008 - for p == 2, 3 (mod 5), b(p^e) = 1 if 4 divides e, and 0 otherwise.
%e A378008 (1 + 1/2^s + 1/3^s + 1/4^s + ...)*(1 + i/2^s - i/3^s - 1/4^s + ...)*(1 - 1/2^s - 1/3^s + 1/4^s + ...)*(1 - i/2^s + i/3^s - 1/4^s + ...) = 1 + 4/11^s + 1/16^s + 4/31^s + 4/41^s + ...
%o A378008 (PARI) A378008(n) = {
%o A378008 my(f = factor(5*n+1), res = 1); for(i=1, #f~,
%o A378008 if(f[i,1] % 5 == 1, res *= binomial(f[i,2]+3, 3));
%o A378008 if(f[i,1] % 5 == 4, if(f[i,2] % 2 == 0, res *= f[i,2]/2+1, return(0)));
%o A378008 if(f[i,1] % 5 == 2 || f[i,1] % 5 == 3, if(f[i,2] % 4 != 0, return(0))));
%o A378008 res; }
%Y A378008 Cf. A378007.
%K A378008 nonn,easy
%O A378008 0,3
%A A378008 _Jianing Song_, Nov 14 2024