A141177 Primes of the form -2*x^2 + 3*x*y + 3*y^2 (as well as of the form 4*x^2 + 7*x*y + y^2).

3, 31, 37, 67, 97, 103, 157, 163, 181, 199, 223, 229, 313, 331, 367, 379, 397, 421, 433, 463, 487, 499, 577, 619, 631, 643, 661, 691, 709, 727, 751, 757, 823, 829, 859, 883, 907, 991, 1021, 1039, 1087, 1093, 1123, 1153, 1171, 1213, 1237, 1279, 1291, 1303, 1321, 1423, 1453, 1483

Offset: 1

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

Keywords

• nonn

Comments

Discriminant = 33. 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.

It is true that A141177(n+1) = A107013(n)? That is: except for p = 3 are these the primes represented by x^2 - x*y + 25*y^2 with x, y nonnegative? - Juan Arias-de-Reyna, Mar 19 2011

From Jianing Song, Jul 30 2018: (Start)

Also primes that are squares modulo 33.

Also primes of the form x^2 - x*y - 8*y^2 with 0 <= x <= y (or x^2 + x*y - 8*y^2 with x, y nonnegative).

These are primes = 3 or congruent to {1, 4, 16, 25, 31} mod 33. Note that the binary quadratic forms with discriminant 33 are in two classes as well as two genera, so there is one class in each genus. A141176 is in the other genus, with primes = 11 or congruent to {2, 8, 17, 29, 32} mod 33.

The observation from Juan Arias-de-Reyna is correct, since the binary quadratic forms with discriminant -99 are also in two classes as well as two genera. Note that -99 = 33*(-3) = (-11)*(-3)^2, so this sequence is essentially the same as A107013.

(End)

Examples

			a(2) = 31 because we can write 31 = -2*4^2 + 3*4*3 + 3*3^2 (or 31 = 4*2^2 + 7*2*1 + 1^2).
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.
• D. B. Zagier, Zetafunktionen und quadratische Körper: Eine Einführung in die höhere Zahlentheorie, Springer-Verlag Berlin Heidelberg, 1981, DOI 10.1007/978-3-642-61829-1.

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A141176 (d=33); A038872 (d=5); A038873 (d=8); A068228, A141123 (d=12); A038883 (d=13); A038889 (d=17); A141111, A141112 (d=65).

Cf. A243185 (numbers of the form -2*x^2 + 3*x*y + 3*y^2).

Cf. A107013.

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

Mathematica

Select[Prime[Range[500]], # == 3 || MatchQ[Mod[#, 33], Alternatives[1, 4, 16, 25, 31]]&] (* Jean-François Alcover, Oct 28 2016 *)