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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 number = 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 (primitive).

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

Both forms [-1, 3, 3] (reduced) and [5, 9, 3] (not reduced) are properly (via a determinant +1 matrix) equivalent to the reduced form [3, 3, -1], a member of the 2-cycle [[3, 3, -1], [-1, 3, 3]]. The other reduced form is the principal form [1, 3, -3], with 2-cycle [[1, 3, -3], [-3, 3, 1]] (see, e.g., A141159, A139492). - Wolfdieter Lang, Jun 24 2019

REFERENCES

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

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

LINKS

Table of n, a(n) for n=1..52.

Peter Luschny, Binary Quadratic Forms

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[250]], MemberQ[{5, 17, 20}, Mod[#, 21]] &]] (* Vincenzo Librandi, Jul 11 2018 *)

PROG

(MAGMA) [3] cat [p: p in PrimesUpTo(2000) | p mod 21 in [5, 17, 20]]; // Vincenzo Librandi, Jul 11 2018

(Sage) # uses[binaryQF]

# The function binaryQF is defined in the link 'Binary Quadratic Forms'.

Q = binaryQF([-1, 3, 3])

Q.represented_positives(1200, 'prime') # Peter Luschny, Jun 24 2019

CROSSREFS

Cf. A141159, A139492 (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: A303839 A148522 A308588 * A336380 A113275 A280080

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 16 00:49 EDT 2021. Contains 343937 sequences. (Running on oeis4.)