cp's OEIS Frontend

A376691 The number of solutions to x + y == x^2 + y^2 == x^3 + y^3 (mod n) with 0 <= x <= y < n.

%I A376691 #19 Oct 02 2024 16:09:40
%S A376691 1,3,3,5,3,10,3,7,5,10,3,16,3,10,10,7,3,18,3,16,10,10,3,24,7,10,5,16,
%T A376691 3,36,3,11,10,10,10,28,3,10,10,24,3,36,3,16,18,10,3,24,9,26,10,16,3,
%U A376691 18,10,24,10,10,3,56,3,10,18,11,10,36,3,16,10,36
%N A376691 The number of solutions to x + y == x^2 + y^2 == x^3 + y^3 (mod n) with 0 <= x <= y < n.
%C A376691 This is the same as the number of solutions to x + y == x^2 + y^2 == x^3+y^3  == x^4 + y^4 (mod n) with x <=y. Proved by Sahaj in Math StackExchange link.
%H A376691 Math StackExchange, <a href="https://math.stackexchange.com/questions/4978600/if-x-y-equiv-x2-y2-equiv-x3-y3-pmodn-is-it-true-that-xy-equ">If x+y == x^2+y^2 == x^3+y^3 (mod n), is it true that x+y==x^4+y^4(mod n)?</a>.
%p A376691 a:=proc(n)
%p A376691  local x,y,count;
%p A376691   count:=0:
%p A376691   for x from 0 to n-1 do
%p A376691    for y from x to n-1 do
%p A376691      if (x+y) mod n =(x^2+y^2) mod n and (x+y) mod n =(x^3+y^3) mod n then count:=count+1;  fi;
%p A376691    od:
%p A376691   od:
%p A376691   count;
%p A376691 end:
%o A376691 (Python)
%o A376691 def A376691(n):
%o A376691     c = 0
%o A376691     for x in range(n):
%o A376691         m = x*(1-x)%n
%o A376691         c += sum(1 for y in range(x,n) if y*(y-1)%n == m and not m*(x-y)%n)
%o A376691     return c # _Chai Wah Wu_, Oct 02 2024
%Y A376691 Cf. A376646.
%K A376691 nonn
%O A376691 1,2
%A A376691 _W. Edwin Clark_, Oct 01 2024