cp's OEIS Frontend

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

%I A359477 #7 Jan 12 2023 01:53:44
%S A359477 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,
%T A359477 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,
%U A359477 4,2,4,4,2,2,2,4,2,4,2,2,2,2,2,2,4,4,4,2,4
%N A359477 a(n) = 2^m(n), where m(n) is the number of distinct primes, neither 2 nor 7, dividing A359476(n).
%C A359477 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].
%C A359477 For details on reduced, primitive forms, proper representations and proper equivalence see A358946 and the two links with references.
%C A359477 The proof runs along the same lines as the one indicated in A358947.
%C A359477 See also the examples in A359476.
%e A359477 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.
%e A359477 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.
%e A359477 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).
%Y A359477 Cf. A358946, A358947, A359476.
%K A359477 nonn
%O A359477 1,1
%A A359477 _Wolfdieter Lang_, Jan 10 2023