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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].

Original entry on oeis.org

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

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Author

Wolfdieter Lang, Jan 10 2023

Keywords

Comments

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.

Examples

			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].

Links

Crossrefs

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

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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.

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

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Jan 10 2023

Keywords

Comments

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.

Examples

			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.

References

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

Crossrefs

Cf. A358946, A359476, A359477.
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