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

0,2

Links

Jianing Song, Table of n, a(n) for n = 0..10000

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.

Example

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 + ...

Prog

(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; }

Crossrefs

Cf. A378007.

Sequence in context: A028625 A344441 A221728 * A339942 A345366 A278720

Adjacent sequences: A378006 A378007 A378008 * A378010 A378011 A378012

Keyword

nonn,easy

Author

Jianing Song, Nov 14 2024

Status

approved