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 A141160 Primes of the form -x^2 + 3*x*y + 3*y^2 (as well as of the form 5*x^2 + 9*x*y + 3*y^2). 6
 3, 5, 17, 41, 47, 59, 83, 89, 101, 131, 167, 173, 227, 251, 257, 269, 293, 311, 353, 383, 419, 461, 467, 479, 503, 509, 521, 563, 587, 593, 647, 677, 719, 761, 773, 797, 839, 857, 881, 887, 929, 941, 971, 983, 1013, 1049, 1091, 1097, 1109, 1151, 1181, 1193 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Discriminant = 21. 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. Except a(1) = 3, primes congruent to {5, 17, 20} mod 21. - Vincenzo Librandi, Jul 11 2018 The comment above is true since the binary quadratic forms with discriminant 21 are in two classes as well as two genera, so there is one class in each genus. A141159 is in the other genus, with primes = 7 or congruent to {1, 4, 16} mod 21. - Jianing Song, Jul 12 2018 4*a(n) can be written in the form 21*w^2 - z^2. - Bruno Berselli, Jul 13 2018 REFERENCES Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966. D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981. LINKS N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references) EXAMPLE a(3)=17 because we can write 17 = -1^2 + 3*1*2 + 3*2^2 (or 17 = 5*1^2 + 9*1*1 + 3*1^2). MATHEMATICA Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 3*x*y + 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *) Join[{3}, Select[Prime[Range], MemberQ[{5, 17, 20}, Mod[#, 21]] &]] (* Vincenzo Librandi, Jul 11 2018 *) PROG (MAGMA)  cat [p: p in PrimesUpTo(2000) | p mod 21 in [5, 17, 20]]; // Vincenzo Librandi, Jul 11 2018 CROSSREFS Cf. A141159 (d=21) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65). Primes in A237351. 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: A148521 A303839 A148522 * A113275 A280080 A001572 Adjacent sequences:  A141157 A141158 A141159 * A141161 A141162 A141163 KEYWORD nonn AUTHOR Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008 EXTENSIONS More terms from Colin Barker, Apr 05 2015 STATUS approved

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Last modified May 25 19:07 EDT 2019. Contains 323576 sequences. (Running on oeis4.)