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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].

%I A359476 #11 Apr 21 2023 12:49:39
%S A359476 3,6,7,14,19,27,31,38,47,54,59,62,63,83,87,94,103,111,118,126,131,139,
%T A359476 159,166,167,171,174,199,203,206,222,223,227,243,251,259,262,271,278,
%U A359476 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
%N 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].
%C A359476 This is a subsequence of A242666.
%C A359476 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.
%C A359476 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.
%C A359476 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.
%C A359476 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.
%e A359476 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.
%e A359476 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.
%e A359476 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.
%Y A359476 Cf. A242666, A358946, A358947, A359477.
%K A359476 nonn
%O A359476 1,1
%A A359476 _Wolfdieter Lang_, Jan 10 2023