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].

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

Author

Wolfdieter Lang, Jan 10 2023

Keywords

• nonn

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.