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

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

Author

Wolfdieter Lang, Jan 10 2023

Keywords

• nonn

Comments

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.

Examples

			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).
		

Crossrefs

Cf. A358946, A358947, A359476.