Primes not of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for nonnegative x, y, and z.
2, 5, 23, 29, 41, 53, 59, 101, 107, 149, 167, 239, 317, 401, 479, 659, 1031, 1409, 1439, 2459, 2543
1
This sequence appears to be finite.
T. D. Noe, Plot of 21 terms
(Mma) nn = 200; c = {1, 1, 1, 1, 1, 1}; t = 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[#] &]; Complement[Prime[Range[PrimePi[t[[-1]]]]], t]
Cf. S000077 (complement in the primes).
nonn
T. D. Noe, Jun 09 2014