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A375003
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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.
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2
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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
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OFFSET
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1,9
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LINKS
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EXAMPLE
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a(9) = 4 since there are solutions (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2).
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PROG
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(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
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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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