cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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).

Original entry on oeis.org

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

Views

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

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

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 *)

A243184 Nonnegative numbers of the form 2*x^2+3*x*y-3*y^2.

Original entry on oeis.org

0, 2, 6, 8, 11, 17, 18, 24, 29, 32, 33, 41, 44, 50, 51, 54, 62, 68, 72, 74, 83, 87, 96, 98, 99, 101, 107, 116, 123, 128, 131, 132, 134, 149, 150, 153, 162, 164, 167, 173, 176, 186, 194, 197, 200, 204, 206, 216, 222, 227, 233, 239, 242, 248, 249, 261, 263, 272, 275, 281, 288, 293, 294, 296, 297
Offset: 1

Views

Author

N. J. A. Sloane, Jun 04 2014

Keywords

Comments

Discriminant 33.

Links

Crossrefs

For primes see A141176. Cf. A243185.

Programs

  • Mathematica
    Reap[For[n = 0, n <= 300, n++, If[Reduce[2*x^2 + 3*x*y - 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]] (* In older versions, this program used to miss out 0 *)
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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