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

Author

T. D. Noe, May 09 2005

Keywords

• nonn
• easy

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

Examples

			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
		

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)

Crossrefs

Cf. A028928.

Programs

Magma

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