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A378012 a(n) = b(10*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 10; 10th column of A378007.

%I A378012 #13 Nov 14 2024 20:13:53
%S A378012 1,4,0,4,4,0,4,4,1,0,4,0,10,4,0,4,0,0,4,4,0,4,0,0,4,4,0,4,4,0,0,4,0,4,
%T A378012 16,0,2,0,0,0,4,0,4,4,0,16,4,0,0,4,0,0,4,0,4,0,0,4,0,0,4,0,0,4,4,0,4,
%U A378012 16,0,4,4,0,0,0,0,4,4,0,16,0,0,4,4,0,2,0,0,0,4,4,0
%N A378012 a(n) = b(10*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 10; 10th column of A378007.
%H A378012 Jianing Song, <a href="/A378012/b378012.txt">Table of n, a(n) for n = 0..10000</a>
%F A378012 a(n) = b(10*n+1), where {b(n)} is multiplicative with:
%F A378012 - b(2^e) = b(5^e) = 0;
%F A378012 - for p == 1 (mod 10), b(p^e) = binomial(e+3,3) = (e+3)*(e+2)*(e+1)/6;
%F A378012 - for p == 9 (mod 10), b(p^e) = e/2 + 1 if e is even, and 0 otherwise;
%F A378012 - for p == 3, 7 (mod 10), b(p^e) = 1 if 4 divides e, and 0 otherwise.
%F A378012 a(n) = A378008(2*n).
%e A378012 (1 + 1/3^s + 1/7^s + 1/9^s + ...)*(1 + i/3^s - i/7^s - 1/9^s + ...)*(1 - 1/3^s - 1/7^s + 1/9^s + ...)*(1 - i/3^s + i/7^s - 1/9^s + ...) = 1 + 4/11^s + 4/31^s + 4/41^s + 4/61^s + 4/71^s + 1/81^s + 4/101^s + ...
%o A378012 (PARI) A378012(n) = {
%o A378012 my(f = factor(10*n+1), res = 1); for(i=1, #f~,
%o A378012 if(f[i,1] % 10 == 1, res *= binomial(f[i,2]+3, 3));
%o A378012 if(f[i,1] % 10 == 9, if(f[i,2] % 2 == 0, res *= f[i,2]/2+1, return(0)));
%o A378012 if(f[i,1] % 10 == 3 || f[i,1] % 10 == 7, if(f[i,2] % 4 != 0, return(0))));
%o A378012 res; }
%K A378012 nonn,easy
%O A378012 0,2
%A A378012 _Jianing Song_, Nov 14 2024