A375003 Number of ordered solutions (x,y,z,w) to x*y + x*z + x*w + y*z + y*w + z*w = n with x,y,z,w >= 1.

0, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 4, 6, 0, 4, 0, 12, 8, 0, 0, 16, 6, 12, 5, 12, 0, 16, 12, 24, 8, 0, 12, 34, 0, 24, 12, 30, 12, 16, 0, 36, 32, 24, 12, 32, 6, 36, 20, 36, 12, 40, 18, 72, 9, 0, 24, 64, 24, 48, 36, 30, 24, 56, 12, 72, 8, 48, 36, 70, 24, 60, 40, 54, 24, 40, 12, 120, 62, 24, 24, 80, 24, 96

Offset: 1

Author

Seiichi Manyama, Jul 27 2024

Keywords

• nonn

Examples

			a(9) = 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. A066851, A374969, A375004.

Programs

PARI

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

Python

from sympy import divisors, integer_nthroot
def A375003(n):
    k = 0
    for c in range(1,n-1):
        for d in divisors(c,generator=True):
            for x in range(1,d):
                xy = x*(d-x)
                a = (c//d)**2
                b = a-(n-c-xy<<2)
                if b>=0:
                    q,r = integer_nthroot(b,2)
                    if r:
                        w = (c//d+q)//2
                        if 1<=wChai Wah Wu, Jul 27 2024