A375004 Number of ordered primitive 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, 4, 12, 0, 16, 12, 24, 8, 0, 12, 34, 0, 24, 8, 30, 12, 16, 0, 36, 32, 24, 12, 32, 6, 36, 16, 36, 12, 40, 12, 72, 8, 0, 24, 64, 24, 48, 32, 30, 24, 56, 12, 72, 8, 48, 24, 70, 24, 60, 32, 54, 24, 40, 12, 120, 62, 24, 24, 76, 24, 96, 32

Offset

1,9

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Prog

(PARI) a(n) = sum(x=1, n, sum(y=1, n, sum(z=1, n, sum(w=1, n, (gcd([x, y, z, w])==1)*(x*y+x*z+x*w+y*z+y*w+z*w==n)))));
(Python)
from math import gcd
from sympy import divisors, integer_nthroot
def A375004(n):
    k = 0
    for c in range(1, n-1):
        for d in divisors(c, generator=True):
            for x in range(1, d):
                y = d-x
                xy = x*y
                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>>1
                        z = c//d-w
                        if 1<=w<c//d and gcd(x, y, z, w)==1:
                            k += 1
                        if q:
                            w = c//d-q>>1
                            z = c//d-w
                            if 1<=w<c//d and gcd(x, y, z, w)==1:
                                k += 1
    return k # Chai Wah Wu, Jul 27 2024

Crossrefs

Cf. A066958, A374970, A375003.

Sequence in context: A204301 A204084 A096904 * A375003 A308257 A308256

Adjacent sequences: A375001 A375002 A375003 * A375005 A375006 A375007

Keyword

nonn

Author

Seiichi Manyama, Jul 27 2024

Status

approved