This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A141188 #38 Oct 12 2022 03:19:40
%S A141188 3,13,17,23,29,43,53,61,79,101,103,107,113,127,131,139,157,173,179,
%T A141188 181,191,199,211,233,251,257,263,269,277,283,311,313,337,347,367,373,
%U A141188 389,419,433,439,443,467,491,503,521,523,547,563,569,571,599,601,607,641
%N A141188 Duplicate of A038883.
%C A141188 Original name was: Primes of the form 3*x^2 + 2*x*y - 4*y^2 (as well as of the form 3*x^2 + 8*x*y + y^2).
%C A141188 Discriminant = 52. Class = 1. 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 A141188 From _Tito Piezas III_, Dec 28 2008: (Start)
%C A141188 Primes of the form u^2 - 13v^2. Using the transformation {u,v} = {4x+y,x} yields the second quadratic form in the title.
%C A141188 This is probably identical to A038883.
%C A141188 (End)
%C A141188 From _Jianing Song_, Sep 20 2018: (Start)
%C A141188 Yes, this is a duplicate of A038883. For primes p congruent to {1, 3, 4, 9, 10, 12} mod 13, they split in the quadratic field Q(sqrt(13)). Since Z[(1+sqrt(13))/2], the ring of integers of Q(sqrt(13)), is a UFD, they are reducible in Z[(1+sqrt(13))/2], so we have p = +-((u' + v'*sqrt(13))/2)*((u' - v'*sqrt(13))/2) = +-(u'^2 - 13*v'^2)/4, where u', v' have the same parity. Since there are elements in Z[(1+sqrt(13))/2] with norm -1 (for example, (3+sqrt(13))/2), we can suppose that p = (u'^2 - 13*v'^2)/4. If u', v' are even, then pick u = u'/2, v = v'/2 and we have p = u^2 - 13*v^2; if u', v' are odd, pick u = (11*u'+-39*v')/4, v = (3*u'+-11*v')/4 by choosing signs appropriately so that u, v are integers.
%C A141188 On the other hand, u^2 - 13*v^2 == 0, 1, 3, 4, 9, 10, 12 (mod 13). So these two sequences are the same.
%C A141188 Also primes of the form x^2 - x*y - 3*y^2 (discriminant 13) with 0 <= x <= y (or x^2 + x*y - 3*y^2 with x, y nonnegative). (End) [Comment partly rewritten by _Jianing Song_, Oct 11 2022]
%e A141188 a(9) = 79 because we can write 79 = 3*5^2 + 2*5*2 - 4*2^2 (or 79 = 3*3^2 + 8*3*2 + 2^2).
%t A141188 Reap[For[p = 2, p < 1000, p = NextPrime[p], If[FindInstance[p == 3*x^2 + 2*x*y - 4*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* _Jean-François Alcover_, Oct 25 2016 *)
%K A141188 dead
%O A141188 1,1
%A A141188 Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008