A141302 Primes of the form -x^2+6*x*y+6*y^2 (as well as of the form 11*x^2+18*x*y+6*y^2).

11, 59, 71, 131, 179, 191, 239, 251, 311, 359, 419, 431, 479, 491, 599, 659, 719, 839, 911, 971, 1019, 1031, 1091, 1151, 1259, 1319, 1439, 1451, 1499, 1511, 1559, 1571, 1619, 1811, 1871, 1931, 1979, 2039, 2099, 2111, 2339, 2351, 2399, 2411, 2459, 2531, 2579, 2591, 2699, 2711

Offset

1,1

Comments

Discriminant = 60. Class number = 4. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1 if they are primitive.

The Pell form X^2 - 15*Y^2 represents the negative primes -a(n), for n >= 1. - Wolfdieter Lang, Nov 28 2024

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..50.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Formula

Primes congruent to {11, 59} (mod 60). -Wolfdieter Lang, Dec 22 2024

Example

a(3)=71 because we can write 71=-1^2+6*1*3+6*3^2 (or 71=11*1^2+18*1*2+6*2^2).

Mathematica

Reap[For[p = 2, p < 3000, p = NextPrime[p], If[FindInstance[p == -x^2 + 6*x*y + 6*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

Crossrefs

Cf. A107152, A141303, A141304 (d=60).

Primes in A237606.

Sequence in context: A356039 A073720 A257364 * A139872 A165977 A214151

Adjacent sequences: A141299 A141300 A141301 * A141303 A141304 A141305

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 24 2008

Extensions

Offset corrected by Mohammed Yaseen, May 20 2023

Status

approved