%I #24 Feb 17 2022 12:00:48
%S 7,19,43,61,73,139,157,163,199,229,271,277,283,313,349,367,397,457,
%T 463,499,541,571,577,613,619,631,643,691,709,727,733,739,757,769,823,
%U 853,859,883,919,937,967,997,1033,1051,1069,1087,1201,1213,1279,1297,1303,1327,1423,1429
%N Primes of the form -3*x^2+3*x*y+4*y^2 (as well as of the form 6*x^2+9*x*y+y^2).
%C Discriminant = 57. 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
%C p = 19 and primes p = 1 mod 3 such that 57 is a square mod p. - _Juan Arias-de-Reyna_, Mar 20 2011
%D Z. I. Borevich and I. R. Shafarevich, Number Theory.
%H Juan Arias-de-Reyna, <a href="/A141193/b141193.txt">Table of n, a(n) for n = 1..10000</a>
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%e a(2)=19 because we can write 19=-3*1^2+3*1*2+4*2^2
%t Select[Prime[Range[250]], # == 19 || MatchQ[Mod[#, 57], Alternatives[1, 4, 7, 16, 25, 28, 43, 49, 55]]&] (* _Jean-François Alcover_, Oct 28 2016 *)
%Y Cf. A141192 (d=57). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
%Y Primes in A243193.
%Y For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%K nonn
%O 1,1
%A Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 24 2008