1,1

Discriminant = 12. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.

This is exactly {2} U A068231, primes congruent to 11 (mod 12). This is because the orders of imaginary quadratic fields with discriminant 12 has 1 class per genus (can be verified by the quadclassunit() function in PARI), so the primes represented by a binary quadratic form of this discriminant are determined by a congruence condition. - Jianing Song, Jun 22 2025

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Robert Israel, Table of n, a(n) for n = 1..6514

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

a(3) = 11 because we can write 11 = -1^2 + 2*1*2 + 2*2^2 (or 11 = 3*1^2 + 6*1*1 + 2*1^2).

N:= 2000:

S:= NULL:

for xx from 1 to floor(2*sqrt(N/3)) do

for yy from ceil(sqrt(max(1, 3*xx^2-N))) to floor(sqrt(3)*xx) do

S:= S, 3*xx^2-yy^2;

od od:

sort(convert(select(isprime, {S}), list)); # Robert Israel, Jul 20 2020

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 2*x*y + 2*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]

(* or: *)

Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* Jean-François Alcover, Oct 25 2016, updated Oct 29 2016 *)

Cf. A038872 (d=5), A038873 (d=8), A068228 (d=12, 48, or -36), A038883 (d=13), A038889 (d=17), A141111 and A141112 (d=65).

Essentially the same as A068231 and A141187.

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Cf. A084917.

Sequence in context: A346605 A091310 A040994 * A263729 A246496 A119641

Adjacent sequences: A141120 A141121 A141122 * A141124 A141125 A141126

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008

More terms from Colin Barker, Apr 05 2015

approved