A358946 -id:A358946 - cp's OEIS frontend

A358947 a(n) = 2^m(n), where m(n) is the number of distinct primes, neither 2 nor 7, dividing A358946(n).

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

Offset: 1

Wolfdieter Lang, Jan 10 2023

nonn

For A358946(1) = 1 one uses m(1) = 0.
a(n) gives the number of representative parallel primitive forms (rpapfs) of Disc = 28 representing k = A358946(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 by F_p = [1, 4, -3].
For details on reduced, primitive or parallel forms, proper representations and proper equivalence see A358946 with the linked papers where references are given.
The proof uses the formula for finding the rpapfs of Disc = 28 representing k, namely c = c(j, k) =(j^2 - 7)/k, for j from {0, 1, ..., k-1}, with the form(s) FPa(k) = [k, 2*j, c(j,k)]. Thus, j^2 - 7 == 0 (mod k).
For the representation k = 1 = A358946(1) there is the unique j = 0 solution with FPa(1) = [1, 0, -7], the neither reduced nor half-reduced Pell form FPell of Disc = 28. FPell is properly equivalent to the reduced F_p.
For k = 2 = A358946(2) the unique solution is j = 1 (-1 == 1 (mod 2)) with FPa(2) = [2, 2,-3], one of the four reduced forms of the principal cycle Cy(28) including F_p as a form. There is no lifting of 2 to powers of 2 (see Apostol, Theorem 5.30, p. 121), hence no factor 2^q, for q >= 2 appears in A358946.
For powers of primes (not 2 or 7) in A358946(n) one has for each prime p two solutions j and p-j of j^2 - 7 == 0 (mod p), and each can be lifted uniquely to powers of this prime. Thus each power of a prime counts as 2, like for the prime. See some examples below.

			k = 57 = 3*19 = A358946(10): One has for k = 3 the two solutions j = 1, 2, giving the parallel forms FPa(3)_1 = [3, 2, -2], belonging to the cycle Cyhat(28), thus 3 is not in A358946 but 3 = A359476(1) for the representaion of k = -3, and FPa(3)_2 = [3, 4, -1], also in Cyhat(28). The lifting of the solutions from k = 3 to k = 3^2 = 9 is possible uniquely because 2*j = 2 and 2*j = 4 are both not congruent to 0 (mod 3). See the two parallel forms for k = 9 above. For k = 19 one has j = 8 and j = 11, with the forms FPa(19)_1 = [19, 16, 3] and FPa(19)_2 = [19, 22, 6], both reaching the cycle Cyhat(28) after R-transformations with t = 3 and first t = -2 then t = 3, respectively. Thus k = 19 belongs to A359476, not A358946.
The four solutions for k = 57 are j = 8, 11, 46, 49, with parallel forms [57, 16, 1], [57, 22, 2], [57, 92, 37] and [57, 98, 42].
The four fundamental representations of F_p = [1, 4, -3] for k = 57 are (11, 16), (5, 4), (6, 7) and (6, 1), respectively.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Theorems 5.28, pp. 118-119, and 5.30, pp. 121-122.

Cf. A358946, A359476, A359477.

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

N. J. A. Sloane, May 31 2014

nonn

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

R. J. Mathar, Table of n, a(n) for n = 0..1184
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes = A141172.

Cf. A242666, A358946.

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

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

A358947 a(n) = 2^m(n), where m(n) is the number of distinct primes, neither 2 nor 7, dividing A358946(n).

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

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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].

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

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