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A376427 The number of distinct values of x+y+z+w (mod n) when x*y*z*w = 1 (mod n).

%I A376427 #18 Jun 02 2025 15:28:08
%S A376427 1,1,3,1,5,3,7,2,5,5,11,3,13,7,15,4,17,5,19,5,21,11,23,6,25,13,15,7,
%T A376427 29,15,31,8,33,17,35,5,37,19,39,10,41,21,43,11,25,23,47,12,49,25,51,
%U A376427 13,53,15,55,14,57,29,59,15,61,31,35,16,65,33,67,17,69,35,71,10,73,37,75,19,77,39,79,20,45,41,83,21,85,43,87,22,89,25
%N A376427 The number of distinct values of x+y+z+w (mod n) when x*y*z*w = 1 (mod n).
%C A376427 The values of n for which a(n) = n seem to agree with A325128. But I have no proof.
%H A376427 Chai Wah Wu, <a href="/A376427/b376427.txt">Table of n, a(n) for n = 1..1688</a>
%p A376427 a:=proc(n)
%p A376427 local x,y,z,w,N;
%p A376427 N:={};
%p A376427 for x from 0 to n-1 do
%p A376427  for y from x to n-1 do
%p A376427   for z from y to n-1 do
%p A376427    for w from z to n-1 do
%p A376427      if (x*y*z*w) mod n = 1 mod n then N:=N union {(x+y+z+w) mod n}; fi;
%p A376427    od:
%p A376427   od:
%p A376427  od:
%p A376427 od:
%p A376427 nops(N);
%p A376427 end:
%o A376427 (Python)
%o A376427 def A376427(n):
%o A376427     s = set()
%o A376427     for x in range(n):
%o A376427         for y in range(x,n):
%o A376427             xy, xyp = x*y%n, (x+y)%n
%o A376427             for z in range(y,n):
%o A376427                 try:
%o A376427                     s.add((xyp+z+pow(xy*z%n,-1,n))%n)
%o A376427                 except:
%o A376427                     continue
%o A376427     return len(s) # _Chai Wah Wu_, Sep 23 2024
%Y A376427 Cf. A376296, A376183, A180783.
%K A376427 nonn
%O A376427 1,3
%A A376427 _W. Edwin Clark_, Sep 22 2024