Numbers n such that x^2 + y^2 (mod n) does not assume every number in Zn, where x and y are nonzero.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 35, 36, 38, 40, 42, 43, 44, 45, 46, 47, 48, 49, 52, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 76, 77, 79, 80, 81, 83, 84, 86, 88, 90, 92, 93, 94, 95

1

T. D. Noe, Plot of terms up to 1000

T. D. Noe, Table of terms up to 1000

(Mma) nn = 100; t = {}; Do[s = Flatten[Table[Mod[x^2 + y^2, n], {x, n-1}, {y, x, n-1}]]; s = Union[s]; If[Length[s] < n, AppendTo[t, n]], {n, nn}]; t

Cf. S000093 (complement).

nonn

T. D. Noe, Jun 23 2014