OFFSET
0,3
LINKS
Jianing Song, Table of n, a(n) for n = 0..10000
FORMULA
a(n) = b(9*n+1), where {b(n)} is multiplicative with:
- b(3^e) = 0;
- for p == 1 (mod 9), b(p^e) = binomial(e+5,5) = (e+5)*(e+4)*(e+3)*(e+2)*(e+1)/120;
- for p == 8 (mod 9), b(p^e) = binomial(e/2+2,2) = (e/2+2)*(e/2+1)/2 if e is even, and 0 otherwise;
- for p == 4, 7 (mod 9), b(p^e) = e/3 + 1 if 3 divides e, and 0 otherwise;
- for p == 2, 5 (mod 9), 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 + 1/4^s + 1/5^s + 1/7^s + 1/8^s + ...)*(1 + (w+1)/2^s + w/4^s - w/5^s - (w+1)/7^s - 1/8^s + ...)*(1 + w/2^s - (w+1)/4^s - (w+1)/5^s + w/7^s + 1/8^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + 1/7^s - 1/8^s + ...)*(1 - (w+1)/2^s + w/4^s + w/5^s - (w+1)/7^s + 1/8^s + ...)*(1 - w/2^s - (w+1)/4^s + (w+1)/5^s + w/7^s - 1/8^s + ...) = 1 + 6/19^s + 6/37^s + 1/64^s + 6/73^s + ...
PROG
(PARI) A378011(n) = {
my(f = factor(9*n+1), res = 1); for(i=1, #f~,
if(f[i, 1] % 9 == 1, res *= binomial(f[i, 2]+5, 5));
if(f[i, 1] % 9 == 8, if(f[i, 2] % 2 == 0, res *= binomial(f[i, 2]/2+2, 2), return(0)));
if(f[i, 1] % 9 == 4 || f[i, 1] % 9 == 7, if(f[i, 2] % 3 == 0, res *= f[i, 2]/3+1, return(0)));
if(f[i, 1] % 9 == 2 || f[i, 1] % 9 == 5, if(f[i, 2] % 6 != 0, return(0))));
res; }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Nov 14 2024
STATUS
approved