cp's OEIS Frontend

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.

%I A375003 #13 Jul 27 2024 12:12:25
%S A375003 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,
%T A375003 12,34,0,24,12,30,12,16,0,36,32,24,12,32,6,36,20,36,12,40,18,72,9,0,
%U A375003 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
%N 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.
%H A375003 Seiichi Manyama, <a href="/A375003/b375003.txt">Table of n, a(n) for n = 1..1000</a>
%e A375003 a(9) = 4 since there are solutions (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2).
%o A375003 (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))));
%o A375003 (Python)
%o A375003 from sympy import divisors, integer_nthroot
%o A375003 def A375003(n):
%o A375003     k = 0
%o A375003     for c in range(1,n-1):
%o A375003         for d in divisors(c,generator=True):
%o A375003             for x in range(1,d):
%o A375003                 xy = x*(d-x)
%o A375003                 a = (c//d)**2
%o A375003                 b = a-(n-c-xy<<2)
%o A375003                 if b>=0:
%o A375003                     q,r = integer_nthroot(b,2)
%o A375003                     if r:
%o A375003                         w = (c//d+q)//2
%o A375003                         if 1<=w<c//d:
%o A375003                             k += 1+bool(q)
%o A375003     return k # _Chai Wah Wu_, Jul 27 2024
%Y A375003 Cf. A066851, A374969, A375004.
%K A375003 nonn
%O A375003 1,9
%A A375003 _Seiichi Manyama_, Jul 27 2024