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

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, 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, 18, 10, 24, 10, 10, 3, 56, 3, 10, 18, 11, 10, 36, 3, 16, 10, 36

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..70.

Math StackExchange, 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)?.

Maple

a:=proc(n)
 local x, y, count;
  count:=0:
  for x from 0 to n-1 do
   for y from x to n-1 do
     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;
   od:
  od:
  count;
end:

Prog

(Python)
def A376691(n):
    c = 0
    for x in range(n):
        m = x*(1-x)%n
        c += sum(1 for y in range(x, n) if y*(y-1)%n == m and not m*(x-y)%n)
    return c # Chai Wah Wu, Oct 02 2024

Crossrefs

Cf. A376646.

Sequence in context: A248955 A071053 A298398 * A094439 A122037 A363796

Adjacent sequences: A376688 A376689 A376690 * A376692 A376693 A376694

Keyword

nonn

Author

W. Edwin Clark, Oct 01 2024

Status

approved