Primes of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for nonnegative x, y, and z.
3, 7, 11, 13, 17, 19, 31, 37, 43, 47, 61, 67, 71, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 137, 139, 151, 157, 163, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 331, 337
1
The complement appears to be finite.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
(Mma) nn = 20; c = {1, 1, 1, 1, 1, 1}; Select[Union[Flatten[Table[c.{x^2, y^2, z^2, x*y, x*z, y*z}, {x, 0, nn}, {y, 0, nn}, {z, 0, nn}]]], # <= nn^2 && PrimeQ[#] &]
Cf. S000078 (complement in the primes).
nonn
T. D. Noe, Jun 09 2014