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 1-3 of 3 results.

A141173 Primes of the form -2*x^2+2*x*y+3*y^2 (as well as of the form 6*x^2+10*x*y+3*y^2).

Original entry on oeis.org

3, 7, 19, 31, 47, 59, 83, 103, 131, 139, 167, 199, 223, 227, 251, 271, 283, 307, 311, 367, 383, 419, 439, 467, 479, 503, 523, 563, 587, 607, 619, 643, 647, 691, 719, 727, 787, 811, 839, 859, 887, 971, 983, 1039, 1063, 1091, 1123, 1151, 1223, 1231, 1259, 1279, 1291, 1307, 1319, 1399, 1427
Offset: 1

Views

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (lourdescm84(AT)hotmail.com), Jun 12 2008

Keywords

Comments

Discriminant = 28. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
Also primes of form 7*u^2-v^2. The transformation {u,v}={-x-y,3*x+2*y} yields the form in the title. [Juan Arias-de-Reyna, Mar 19 2011]
This is also the list of primes p such that p = 7 or p is congruent to 3, 19 or 27 mod 28. - Jean-François Alcover, Oct 28 2016

Examples

			a(3)=19 because we can write 19=-2*4^2+2*4*3+3*3^2 (or 19=6*1^2+10*1*1+3*1^2).

References

  • Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

Crossrefs

Cf. A141172 (d=28) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).
Cf. also A242666.
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[250]], # == 7 || MatchQ[Mod[#, 28], 3|19|27]&] (* Jean-François Alcover, Oct 28 2016 *)

A242662 Nonnegative integers of the form x^2 + 4xy - 3y^2.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 16, 18, 21, 25, 29, 32, 36, 37, 42, 49, 50, 53, 57, 58, 64, 72, 74, 81, 84, 93, 98, 100, 106, 109, 113, 114, 116, 121, 128, 133, 137, 141, 144, 148, 149, 162, 168, 169, 177, 186, 189, 193, 196, 197, 200, 212, 217, 218, 225, 226, 228, 232, 233, 242, 249, 256, 261, 266, 274, 277, 281, 282, 288, 289, 296, 298
Offset: 0

Views

Author

N. J. A. Sloane, May 31 2014

Keywords

Comments

Discriminant = 28.
Also nonnegative integers of the form x^2 - 7y^2. - Colin Barker, Sep 29 2014
Also nonnegative integers of the form x^2 + bxy + cy^2 where b = -2n, c = n^2 - 7, for integer n. This includes both forms above: x^2 + 4xy - 3y^2 with n = -2 and x^2 - 7y^2 with n = 0. - Klaus Purath, Jan 14 2023
For the subsequence of numbers that are properly represented see A358946. - Wolfdieter Lang, Jan 18 2023
Proof for the proper equivalence of the above given family of forms F(n) = [1, -2*n, n^2 -7], for integer n, with the reduced principal form of discriminant 28, namely F_p = [1, 4, -3] given in the name: In matrix form MF(n) = Matrix([[1, -n], [-n, n^2 -7]]) = R(n)^T*MF_p(n)*R(n), with MF_p(n) = Matrix([[1, 2], [2, -3]]) and R(n) = Matrix([[1, -(n+2)], [0, 1]]) (T for transposed). - Wolfdieter Lang, Jan 20 2023

Links

Crossrefs

Primes = A141172.
Cf. A242666, A358946.

Programs

  • Mathematica
    Reap[For[n = 0, n <= 300, n++, If[Reduce[x^2 + 4*x*y - 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

A359476 The sequence {-a(n)}_{n>=1} gives all negative integers that are properly represented by each primitive binary quadratic forms of discriminant 28 that is properly equivalent to the reduced principal form [1, 4, -3].

Original entry on oeis.org

3, 6, 7, 14, 19, 27, 31, 38, 47, 54, 59, 62, 63, 83, 87, 94, 103, 111, 118, 126, 131, 139, 159, 166, 167, 171, 174, 199, 203, 206, 222, 223, 227, 243, 251, 259, 262, 271, 278, 279, 283, 307, 311, 318, 327, 334, 339, 342, 367, 371, 383, 398, 399, 406, 411, 419, 423, 439, 446, 447, 454, 467, 479, 486
Offset: 1

Views

Author

Wolfdieter Lang, Jan 10 2023

Keywords

Comments

This is a subsequence of A242666.
For details on indefinite binary quadratic primitive forms F = a*x^2 + b*x*y + c*y^2 (gcd(a, b, c) = 1), also denoted by F = [a, b, c], with discriminant Disc = b^2 - 4*a*c = 28 = 2^2*7, see A358946 and A358947.
Each primitive form, properly equivalent to the reduced principal form F_p = [1, 4, -3] for Disc = 28 (used in -A242666), represents the given negative k = -a(n) values (and only these) properly with X = (x, y), i.e., gcd(x, y) = 1. Modulo an overall sign change in X one can choose x nonnegative.
There are A359477(n) representative parallel primitive forms (rpapfs) of discriminant Disc = 28 for k = -a(n). This gives the number of proper fundamental representations (x, y), with x >= 0, of each primitive form [a, b, c], properly equivalent to the principal form F_p of Disc = 28.
For the positive integers k, properly represented by primitive forms [a, b, c] which are properly equivalent to the principal form F_p for Disc = 28, see A358946. The corresponding number of fundamental proper representations is given in A358947.

Examples

			k = -a(1) = -3: the 2 = A359477(1) representative parallel primitive forms (rpapfs) for Disc = 28 are [-3, 2, 2] and, [-3, 4, 1]. See the examples in A358947 for k = 57 = 3*19, and for the fundamental representations see A359477.
k = -a(3) = -7: The 1 = A359477(3) rpapf for Disc = 28 is [-7, 0, 1]. See a comment in A358947 for k = 7, and A359477.
k = -a(15) = -87: The 4 = A359477(15) rpapfs for Disc = 28 are [-87, 46, -6], [-87, 70, -14], [-87, 104, -31], and [-87, 128, -47]. See A359477 for the fundamental representations.

Crossrefs

Cf. A242666, A358946, A358947, A359477.
Showing 1-3 of 3 results.

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

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