A107006 Primes of the form 4x^2-4xy+7y^2, with x and y nonnegative.

7, 31, 79, 103, 127, 151, 199, 223, 271, 367, 439, 463, 487, 607, 631, 727, 751, 823, 919, 967, 991, 1039, 1063, 1087, 1231, 1279, 1303, 1327, 1399, 1423, 1447, 1471, 1543, 1567, 1663, 1759, 1783, 1831, 1879, 1951, 1999, 2143, 2239, 2287, 2311

Offset: 1

Author

T. D. Noe, May 09 2005

Keywords

• nonn
• easy

Comments

Discriminant=-96.

Also, primes of the form 24n+7. - Artur Jasinski, Nov 25 2007 [See the Reble link]

Also primes of the forms 4x^2+4xy+7y^2, 7x^2+6xy+15y^2, 7x^2+2xy+7y^2 and 7x^2+4xy+28y^2. See A140633. - T. D. Noe, May 19 2008

Also, primes of form u^2+6v^2 with odd v while sequence A107008 is even v. This can be seen by expressing its form as (2x-y)^2+6y^2 (where y can only be odd) while the latter is x^2+6(2y)^2. Additionally, this sequence is 7 mod 24 while the second is 1 mod 24 and together, they are the primes of form x^2+6y^2 (A033199) which are either {1,7} mod 24. - Tito Piezas III, Jan 01 2009

Links

• Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 168 terms from N. J. A. Sloane]
• Don Reble, Notes on this sequence
• J. Liouville, Théorème concernant les nombres premiers de la forme 24µ + 7, Journal de mathématiques pures et appliquées 2e série, tome 4 (1859), pp. 399-400.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A124477.

Programs

Mathematica

a = {}; Do[If[PrimeQ[24n + 7], AppendTo[a, 24n + 7]], {n, 0, 100}]; a (* Artur Jasinski, Nov 25 2007 *)
QuadPrimes2[4, -4, 7, 10000] (* see A106856 *)
Select[24*Range[0,4000]+7,PrimeQ] (* Harvey P. Dale, May 13 2018 *)

Extensions

Recomputed b-file and deleted erroneous Mma program by N. J. A. Sloane, Jun 08 2014