A106915 Primes of the form 3x^2 + 2xy + 5y^2, with x and y any integer.

3, 5, 13, 19, 59, 61, 83, 101, 131, 139, 157, 173, 181, 227, 229, 251, 269, 283, 293, 307, 349, 397, 419, 461, 467, 509, 523, 563, 587, 619, 643, 661, 677, 691, 733, 773, 787, 797, 811, 829, 853, 859, 941, 971, 997, 1013, 1021, 1069, 1091, 1109, 1123

Offset

1,1

Comments

Discriminant = -56.

Also primes congruent to {3,5,13,19,27,45} mod 56. - Vincenzo Librandi, Jul 02 2016

The theta series for the quadratic form 3x^2 + 2xy + 5y^2 is the g.f. of A028928. - Michael Somos, Jul 02 2016

Legendre symbol (-14, a(n)) = Kronecker symbol (a(n), 14) = 1. Also, this sequence lists primes p such that Kronecker symbol (p, 2) = Legendre symbol (p, 7) = -1, i.e., primes p == 3, 5 (mod 8) and 3, 5, 6 (mod 7). - Jianing Song, Sep 04 2018

Links

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Example

59 is in the sequence since it is prime, and 59 = 3x^2 + 2xy + 5y^2 with x = 3 and y = 2. - Michael B. Porter, Jul 02 2016

Mathematica

Union[QuadPrimes2[3, 2, 5, 10000], QuadPrimes2[3, -2, 5, 10000]] (* see A106856 *)
Select[Prime@Range[600], MemberQ[{3, 5, 13, 19, 27, 45}, Mod[#, 56]] &] (* Vincenzo Librandi, Jul 02 2016 *)

Prog

(Magma) [p: p in PrimesUpTo(3000) | p mod 56 in {3, 5, 13, 19, 27, 45}]; // Vincenzo Librandi, Jul 02 2016

Crossrefs

Cf. A028928.

Sequence in context: A141215 A191039 A240278 * A112928 A106916 A249340

Adjacent sequences: A106912 A106913 A106914 * A106916 A106917 A106918

Keyword

nonn,easy

Author

T. D. Noe, May 09 2005

Status

approved