A358947 -id:A358947 - cp's OEIS frontend

A358946 Positive integers that are properly represented by each primitive binary quadratic form of discriminant 28 that is properly equivalent to the principal form [1, 4, -3].

1, 2, 9, 18, 21, 29, 37, 42, 53, 57, 58, 74, 81, 93, 106, 109, 113, 114, 133, 137, 141, 149, 162, 177, 186, 189, 193, 197, 217, 218, 226, 233, 249, 261, 266, 274, 277, 281, 282, 298, 309, 317, 329, 333, 337, 354, 361, 373, 378, 386, 389, 393, 394, 401, 413, 417, 421, 434, 449, 457, 466, 477, 498, 501

Offset: 1

Wolfdieter Lang, Jan 10 2023

nonn

This is a subsequence of A242662, excluding the primitive forms of discriminant 28 with only improper representations of k, like k = 4, 8, 16, 25, 32, ... .
An indefinite binary quadratic primitive form F = a*x^2 + b*x*y + c*y^2 (gcd(a, b, c) = 1) with discriminant Disc = b^2 - 4*a*c = 28 = 2^2*7 is denoted by [a, b, c], or in matrix notation by MF = Matrix([[a, b/2], [b/2, c]]). Hence F = X*MF*X^T (T for transposed), where X = (x, y). See the two links for details and references.
Properly equivalent forms F' and F are related by a matrix R of determinant +1 like MF' = R^T*MF*R, and X'^T = R^{-1}*X^T.
Each primitive form, properly equivalent to the reduced principal form F_p = [1, 4, -3] for Disc = 28 (used in A242662), represents the given nonnegative k = a(n) values (and only these) properly with X = (x, y) and gcd(x, y) = 1. Modulo an overall sign change in X one can choose x nonnegative.
There are 8 = A082174(8) primitive reduced forms of Disc = 28 leading to 2 = A087048(8) (class number) cycles each of period 4, namely the principal cycle CyP = [[1, 4, -3], [-3, 2, 2], [2, 2, -3], [-3, 4, 1]] and the one (with outer signs flipped) CyP' = [[-1, 4, 3], [3, 2, -2], [-2, 2, 3], [3, 4, -1]].
There are A358947(n) representative parallel primitive forms (rpapfs) of discriminant Disc = 28 for k = a(n). This gives the number of proper fundamental representations X = (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 negative integers k properly represented by primitive forms [a, b, c] properly equivalent to the principal form of Disc = 28 see A359476. The corresponding number of fundamental proper representations is given in A359477.
This and the three related sequences originated from a proposal by Klaus Purath proving that the form FKP := [1, -2, -6] of Disc = 28 represents k = k(m) = m^2 - 7 = A028881(m), for m >= 3, with the two fundamental representations X1(m) = (m+1, 1) and X2(m) = (11*m - 29, 3*m - 8). This form FKP is properly equivalent to the principal form F_p with R = Matrix([[1, -3], [0, 1]]). Hence all k = a(n) are represented by the form FKP, and A028881 is a subsequence of the present one.

			k = 9 = a(3): F = FPell = [1, 0, -7] is properly equivalent to F_p = [1, 4, -3] by two so-called half-reduced right neighbor R(t)-transformations, with the matrix R = R(t) = Matrix([[0, -1], [1, t]]), first with t = 0 then with t = 2. For FPell representing k = 9 with x > 0 and y > 0 see X_1(9, i) = (A307168(i), A307169(i)) and X_2(9, i) = (A307172(i), A307173(i)), for i >= 0. There are also the representations with y -> -y arising from the opposite fundamental solutions.
The 2 = A358947(3) rpapfs are F1 = [9, 8, 1] and F2 = [9, 10, 2]. They lead by proper equivalence transformations to a form of the above given principal cycle CyP. F1 -> [1, 4, -3] = F_p with matrix R(6), and F2 -> [2, 2, -3] with R(3). See the FIGURE, p. 10, of the linked paper.
Besides the primitive forms FPell, F1, F2 and the four forms of CyP also F' = [-7, 0, 1], and all primitive and properly equivalent forms represent k = 9. See the mentioned FIGURE, where FPa1 = F1, FPa1 = F2, Fpa2' = F_p^{(2)} = [2, 2, -3] and FPa2'' = F_p^{(3)} = [-3, 4, 1].

Wolfdieter Lang, Periods of Indefinite Binary Quadratic Forms, Continued Fractions and the Pell +/-4 Equations.
Wolfdieter Lang, Cycles of reduced Pell forms, general Pell equations and Pell graphs

Cf. A028881, A242662, A307168, A307169, A307172, A307173, A358947, A359476, A359477.

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

Wolfdieter Lang, Jan 10 2023

nonn

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.

			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.

Cf. A242666, A358946, A358947, A359477.

A359477 a(n) = 2^m(n), where m(n) is the number of distinct primes, neither 2 nor 7, dividing A359476(n).

Original entry on oeis.org

2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 2, 4, 4, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 4, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 4, 4, 4, 2, 4

Offset: 1

Wolfdieter Lang, Jan 10 2023

nonn

a(n) gives the number of representative parallel primitive forms (rpapfs) of Disc = 28 representing k = -A359476(n), that is the number of proper fundamental representations X = (x, y) of each indefinite primitive binary quadratic form of discriminant Disc = 28 = 2^2*4 which is properly equivalent to  the reduced principal form F_p = x^2 + 4*x*y - 3*y^2, denoted also by F_p = [1, 4, -3].
For details on reduced, primitive forms, proper representations and proper equivalence see A358946 and the two links with references.
The proof runs along the same lines as the one indicated in A358947.
See also the examples in A359476.

			k = -A359476(1) = -3: The 2 = a(1) proper fundamental representation of F_p = [1, 4, -3], from the two rpapfs given in the example of A359476, are  X(-3)_1 = (0, 1) and  X(-3)_2 = (1, 2), respectively. The first result uses the transformation R(-1) (for R(t) see the Pell example in A358946) acting on the trivial solution (1, 0)^T (T for transposed) of the first rpapf. For the second result R^{-1}(4) (1, 0)^T = (4, -1), which becomes (-1, -2) after applying the automorphic matrix Auto(28) = Matrix([[2,9],[3,14]]) for the 4-cycle of Disc = 28, and this is replaced by (1, 2) with x >= 0.
k = -A359476(3) = -7: The 1 = a(3) rpapf [-7, 0, 1] leads to the proper fundamental solution X(-7) = (2, -1), after applying R^{-1}(2) on (1, 0)^T.
k = -A359476(15) = -87: The 4 = a(15) rpapfs given in A359476 lead to the proper fundamental representation X(-87)_1 = (10, 17), X(-87)_2 = (2, 7), X(-87)_3 = (3, 8), and X(-87)_4 = (3, -4).

Cf. A358946, A358947, A359476.

cp's OEIS Frontend

A358946 Positive integers that are properly represented by each primitive binary quadratic form of discriminant 28 that is properly equivalent to the principal form [1, 4, -3].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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].

Views

Author

Keywords

Comments

Examples

Crossrefs

A359477 a(n) = 2^m(n), where m(n) is the number of distinct primes, neither 2 nor 7, dividing A359476(n).

Views

Author

Keywords

Comments

Examples

Crossrefs