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.
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, 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, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, 6, 0, 6
Offset: 0
Examples
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 + ...
Links
- Jianing Song, Table of n, a(n) for n = 0..10000
Crossrefs
Cf. A378007.
Programs
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PARI
A378009(n) = { my(f = factor(7*n+1), res = 1); for(i=1, #f~, if(f[i,1] % 7 == 1, res *= binomial(f[i,2]+5, 5)); if(f[i,1] % 7 == 6, if(f[i,2] % 2 == 0, res *= binomial(f[i,2]/2+2, 2), return(0))); 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))); if(f[i,1] % 7 == 3 || f[i,1] % 7 == 5, if(f[i,2] % 6 != 0, return(0)))); res; }
Formula
a(n) = b(7*n+1), where {b(n)} is multiplicative with:
- b(7^e) = 0;
- for p == 1 (mod 7), b(p^e) = binomial(e+5,5) = (e+5)*(e+4)*(e+3)*(e+2)*(e+1)/120;
- 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;
- for p == 2, 4 (mod 7), b(p^e) = e/3 + 1 if 3 divides e, and 0 otherwise;
- for p == 3, 5 (mod 7), b(p^e) = 1 if 6 divides e, and 0 otherwise.