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A378009 a(n) = b(7*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 7; 7th column of A378007.

%I A378009 #12 Nov 14 2024 23:23:19
%S A378009 1,2,0,0,6,0,6,0,0,3,6,0,0,0,0,0,6,0,6,0,0,0,0,0,3,0,0,0,6,0,6,0,0,12,
%T A378009 6,0,0,0,0,0,6,0,0,0,0,0,0,0,6,12,0,0,0,0,6,0,0,0,0,0,6,0,0,0,6,0,6,0,
%U A378009 0,0,6,0,0,4,0,0,0,0,6,0,0,12,0,0,0,0,0,0,6,0,6
%N A378009 a(n) = b(7*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 7; 7th column of A378007.
%H A378009 Jianing Song, <a href="/A378009/b378009.txt">Table of n, a(n) for n = 0..10000</a>
%F A378009 a(n) = b(7*n+1), where {b(n)} is multiplicative with:
%F A378009 - b(7^e) = 0;
%F A378009 - for p == 1 (mod 7), b(p^e) = binomial(e+5,5) = (e+5)*(e+4)*(e+3)*(e+2)*(e+1)/120;
%F A378009 - for p == 6 (mod 7), b(p^e) = binomial(e/2+2,2) = (e/2+2)*(e/2+1)/2 if e is even, and 0 otherwise;
%F A378009 - for p == 2, 4 (mod 7), b(p^e) = e/3 + 1 if 3 divides e, and 0 otherwise;
%F A378009 - for p == 3, 5 (mod 7), b(p^e) = 1 if 6 divides e, and 0 otherwise.
%e A378009 Write w = exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2, then (1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + ...)*(1 + w/2^s + (w+1)/3^s - (w+1)/4^s - w/5^s - 1/6^s + ...)*(1 - (w+1)/2^s + w/3^s + w/4^s - (w+1)/5^s + 1/6^s + ...)*(1 + 1/2^s - 1/3^s + 1/4^s - 1/5^s - 1/6^s + ...)*(1 + w/2^s - (w+1)/3^s - (w+1)/4^s + w/5^s + 1/6^s + ...)*(1 - (w+1)/2^s - w/3^s + w/4^s + (w+1)/5^s - 1/6^s + ...) = 1 + 2/8^s + 6/29^s + 6/43^s + 3/64^s + 6/71^s + ...
%o A378009 (PARI) A378009(n) = {
%o A378009 my(f = factor(7*n+1), res = 1); for(i=1, #f~,
%o A378009 if(f[i,1] % 7 == 1, res *= binomial(f[i,2]+5, 5));
%o A378009 if(f[i,1] % 7 == 6, if(f[i,2] % 2 == 0, res *= binomial(f[i,2]/2+2, 2), return(0)));
%o A378009 if(f[i,1] % 7 == 2 || f[i,1] % 7 == 4, if(f[i,2] % 3 == 0, res *= f[i,2]/3+1, return(0)));
%o A378009 if(f[i,1] % 7 == 3 || f[i,1] % 7 == 5, if(f[i,2] % 6 != 0, return(0))));
%o A378009 res; }
%Y A378009 Cf. A378007.
%K A378009 nonn,easy
%O A378009 0,2
%A A378009 _Jianing Song_, Nov 14 2024