A374969 Number of ordered solutions (x,y,z,w) to x*y + y*z + z*w + w*x = n with x,y,z,w >= 1.

0, 0, 0, 1, 0, 4, 0, 6, 4, 8, 0, 22, 0, 12, 16, 23, 0, 36, 0, 42, 24, 20, 0, 80, 16, 24, 32, 62, 0, 104, 0, 72, 40, 32, 48, 151, 0, 36, 48, 148, 0, 152, 0, 102, 120, 44, 0, 242, 36, 120, 64, 122, 0, 200, 80, 216, 72, 56, 0, 396, 0, 60, 176, 201, 96, 248, 0, 162, 88, 280, 0, 486, 0, 72, 208, 182

Offset: 1

Author

Seiichi Manyama, Jul 26 2024

Keywords

• nonn

Comments

a(n) = 0 if and only if n = 1 or n is prime. - Chai Wah Wu, Jul 26 2024

Examples

			a(6) = 4 since there are solutions (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A000203, A066851, A374970.

Programs

PARI

a(n) = sum(x=1, n, sum(y=1, n, sum(z=1, n, sum(w=1, n, x*y+y*z+z*w+w*x==n))));

Python

from math import prod
from sympy import factorint
def A374969(n):
    f = factorint(n).items()
    return (n+1)*prod(e+1 for p,e in f)-(prod((p**(e+1)-1)//(p-1) for p,e in f)<<1) # Chai Wah Wu, Jul 26 2024

Formula

a(n) = (n+1)*A000005(n)-2*A000203(n). - Chai Wah Wu, Jul 26 2024