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A376296 The number of solutions x<=y<=z<=w in Z/(n) of the equation x+y+z+w = x*y*z*w.

%I A376296 #31 Oct 01 2024 12:19:42
%S A376296 1,2,6,7,14,18,27,34,51,59,91,96,134,136,208,203,285,261,385,373,493,
%T A376296 487,650,616,818,750,949,947,1240,1146,1517,1397,1766,1662,2089,1824,
%U A376296 2443,2309,2723,2638,3311,2977,3801,3482,4024,3962,4900,4382,5525,5023,6078
%N A376296 The number of solutions x<=y<=z<=w in Z/(n) of the equation x+y+z+w = x*y*z*w.
%H A376296 Chai Wah Wu, <a href="/A376296/b376296.txt">Table of n, a(n) for n = 1..1351</a>
%p A376296 a:=proc(n)
%p A376296 local x,y,z,w,N;
%p A376296 N:=0:
%p A376296 for x from 0 to n-1 do
%p A376296  for y from x to n-1 do
%p A376296   for z from y to n-1 do
%p A376296    for w from z to n-1 do
%p A376296     if (x+y+z+w-x*y*z*w) mod n = 0 then N:=N + 1; fi;
%p A376296    od:
%p A376296   od:
%p A376296  od:
%p A376296 od:
%p A376296 N;
%p A376296 end:
%o A376296 (Python)
%o A376296 def A376296(n):
%o A376296     c = 0
%o A376296     for x in range(n):
%o A376296         for y in range(x,n):
%o A376296             xy,xyp = x*y%n,(x+y)%n
%o A376296             for z in range(y,n):
%o A376296                 xyz, xyzp = xy*z%n-1,(xyp+z)%n
%o A376296                 c += sum(not (xyz*w-xyzp)%n for w in range(z,n))
%o A376296     return c # _Chai Wah Wu_, Sep 19 2024
%Y A376296 Cf. A376183, A180783.
%K A376296 nonn
%O A376296 1,2
%A A376296 _W. Edwin Clark_, Sep 19 2024