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A141170
Primes of the form x^2+4*x*y-2*y^2 (as well as of the form 3*x^2+6*x*y+y^2).
7
3, 19, 43, 67, 73, 97, 139, 163, 193, 211, 241, 283, 307, 313, 331, 337, 379, 409, 433, 457, 499, 523, 547, 571, 577, 601, 619, 643, 673, 691, 739, 769, 787, 811, 859, 883, 907, 937, 1009, 1033, 1051, 1123, 1129, 1153, 1171, 1201, 1249, 1291, 1297, 1321, 1459, 1483, 1489, 1531
OFFSET
1,1
COMMENTS
Discriminant = 24. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
Also, primes of form u^2 - 6v^2. The transformation {u,v} = {x+2y,y} yields the form in the title. - Tito Piezas III, Dec 31 2008
Conjecture: this is also the list of primes that are simultaneously of the form x^2+2y^2 and of the form x^2+3y^2; that is, the intersection of A002476 and A033203. - Zak Seidov, Jun 07 2014
This is also the list of primes p such that p = 3 or p is congruent to 1 or 19 mod 24. - Jean-François Alcover, Oct 28 2016
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
LINKS
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
EXAMPLE
a(2)=19 because we can write 19=3^2+4*3*1-2*1^2 (or 19=3*1^2+6*1*2+2^2)
MATHEMATICA
xy[{x_, y_}]:={x^2 + 4 x y - 2 y^2, y^2 + 4 x y - 2 x^2}; Union[Select[Flatten[xy/@Subsets[Range[40], {2}]], #>0&&PrimeQ[#]&]] (* Vincenzo Librandi, Jun 09 2014 *)
Select[Prime[Range[250]], # == 3 || MatchQ[Mod[#, 24], 1|19]&] (* Jean-François Alcover, Oct 28 2016 *)
CROSSREFS
Cf. A141171 (d=24), A106950 (Primes of the form x^2+18y^2), A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Cf. also A242661, A002476, A033203.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Sequence in context: A227878 A042371 A141644 * A107154 A141373 A336792
KEYWORD
nonn
AUTHOR
Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008
STATUS
approved