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.

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, 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, 0, 0, 0, 4, 4, 0, 4, 4, 0, 4, 3, 0, 4, 0, 8, 0, 4, 0, 0, 16, 0, 0, 0

Offset

0,2

Links

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

Formula

a(n) = b(8*n+1), where {b(n)} is multiplicative with:

- b(2^e) = 0;

- for p == 1 (mod 8), b(p^e) = binomial(e+3,3) = (e+3)*(e+2)*(e+1)/6;

- for p == 3, 5, 7 (mod 8), b(p^e) = e/2 + 1 if e is even, and 0 otherwise.

Example

(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 + ...

Prog

(PARI) A378010(n) = {
my(f = factor(8*n+1), res = 1); for(i=1, #f~,
if(f[i, 1] % 8 == 1, res *= binomial(f[i, 2]+3, 3));
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))));
res; }

Crossrefs

Cf. A378007.

Sequence in context: A331144 A258711 A127278 * A356165 A366589 A335764

Adjacent sequences: A378007 A378008 A378009 * A378011 A378012 A378013

Keyword

nonn,easy

Author

Jianing Song, Nov 14 2024

Status

approved