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.

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, 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, 16, 0, 4, 4, 0, 0, 0, 0, 4, 4, 0, 16, 0, 0, 4, 4, 0, 2, 0, 0, 0, 4, 4, 0

Offset: 0

Author

Jianing Song, Nov 14 2024

Keywords

• nonn
• easy

Examples

			(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 + ...
		

Links

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

Programs

PARI

A378012(n) = {
my(f = factor(10*n+1), res = 1); for(i=1, #f~,
if(f[i,1] % 10 == 1, res *= binomial(f[i,2]+3, 3));
if(f[i,1] % 10 == 9, if(f[i,2] % 2 == 0, res *= f[i,2]/2+1, return(0)));
if(f[i,1] % 10 == 3 || f[i,1] % 10 == 7, if(f[i,2] % 4 != 0, return(0))));
res; }

Formula

a(n) = b(10*n+1), where {b(n)} is multiplicative with:

- b(2^e) = b(5^e) = 0;

- for p == 1 (mod 10), b(p^e) = binomial(e+3,3) = (e+3)*(e+2)*(e+1)/6;

- for p == 9 (mod 10), b(p^e) = e/2 + 1 if e is even, and 0 otherwise;

- for p == 3, 7 (mod 10), b(p^e) = 1 if 4 divides e, and 0 otherwise.

a(n) = A378008(2*n).