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A378010 a(n) = b(8*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 8; 8th column of A378007.

%I A378010 #12 Nov 14 2024 23:23:14
%S A378010 1,2,4,2,0,4,2,0,0,4,3,4,4,0,4,2,0,4,0,8,0,2,0,0,4,0,0,0,4,4,4,0,4,0,
%T A378010 0,4,10,0,0,4,0,0,4,0,4,2,8,0,0,0,4,4,0,8,4,4,4,4,0,0,0,0,0,0,0,4,2,0,
%U A378010 0,0,0,4,4,0,4,4,0,4,3,0,4,0,8,0,4,0,0,16,0,0,0
%N A378010 a(n) = b(8*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 8; 8th column of A378007.
%H A378010 Jianing Song, <a href="/A378010/b378010.txt">Table of n, a(n) for n = 0..10000</a>
%F A378010 a(n) = b(8*n+1), where {b(n)} is multiplicative with:
%F A378010 - b(2^e) = 0;
%F A378010 - for p == 1 (mod 8), b(p^e) = binomial(e+3,3) = (e+3)*(e+2)*(e+1)/6;
%F A378010 - for p == 3, 5, 7 (mod 8), b(p^e) = e/2 + 1 if e is even, and 0 otherwise.
%e A378010 (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 + 1/3^s - 1/5^s - 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...)*(1 - 1/3^s - 1/5^s + 1/7^s + ...) = 1 + 2/9^s + 4/17^s + 2/25^s + 4/41^s + 2/49^s + 4/73^s + 3/81^s + ...
%o A378010 (PARI) A378010(n) = {
%o A378010 my(f = factor(8*n+1), res = 1); for(i=1, #f~,
%o A378010 if(f[i,1] % 8 == 1, res *= binomial(f[i,2]+3, 3));
%o A378010 if(f[i,1] % 8 == 3 || f[i,1] % 8 == 5 || f[i,1] % 8 == 7, if(f[i,2] % 2 == 0, res *= f[i,2]/2+1, return(0))));
%o A378010 res; }
%Y A378010 Cf. A378007.
%K A378010 nonn,easy
%O A378010 0,2
%A A378010 _Jianing Song_, Nov 14 2024