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

1,9

Links

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

Example

a(9) = 4 since there are solutions (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2).

Prog

(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<=w<c//d:
                            k += 1+bool(q)
    return k # Chai Wah Wu, Jul 27 2024

Crossrefs

Cf. A066851, A374969, A375004.

Sequence in context: A204084 A096904 A375004 * A308257 A308256 A096406

Adjacent sequences: A375000 A375001 A375002 * A375004 A375005 A375006

Keyword

nonn

Author

Seiichi Manyama, Jul 27 2024

Status

approved