Primes of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for positive x, y, and z.
11, 17, 43, 47, 61, 67, 71, 73, 83, 89, 97, 103, 109, 113, 131, 137, 139, 151, 157, 163, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383
1
The complement in the primes appears to be finite. Note that sequence A042998, primes of the form x^2 + y^2 + z^2, is missing far more primes. The discriminant of a third-order quadratic form is a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z is a*f^2 + b*e^2 + c*d^2 - 4*a*b*c - d*e*f. Hence, the discriminant of this quadratic form is -2.
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, nn}, {y, nn}, {z, nn}]]], # <= nn^2 && PrimeQ[#] &]
Cf. S000076 (complement in the primes).
nonn
T. D. Noe, Jun 09 2014