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

0,2

Links

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

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).

Example

(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 + ...

Prog

(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; }

Crossrefs

Sequence in context: A062524 A152856 A031362 * A198414 A377784 A309721

Adjacent sequences: A378009 A378010 A378011 * A378013 A378014 A378015

Keyword

nonn,easy

Author

Jianing Song, Nov 14 2024

Status

approved