cp's OEIS Frontend

A256945 Number of periods of reduced indefinite binary quadratic forms with discriminant D(n) = A079896(n).

%I A256945 #30 Jun 01 2025 10:04:40
%S A256945 1,1,2,1,1,2,2,2,2,1,3,2,1,2,1,2,3,4,2,1,2,2,4,1,2,2,2,3,1,2,2,4,4,2,
%T A256945 2,1,2,2,6,1,1,2,4,4,1,4,1,2,3,4,2,2,5,2,4,2,4,1,4,2,4,4,1,2,3,4,1,6,
%U A256945 2,2,4,4,2,1,4,2,6,1,2,2,2,4,8,1,1,3,2,4,4,4,2,2,2,4,2,4
%N A256945 Number of periods of reduced indefinite binary quadratic forms with discriminant D(n) = A079896(n).
%C A256945 This is an ``imprimitive'' class number.  Each a(n) is A087048(n) increased by the number of cycles of discriminant D(n) of imprimitive binary quadratic forms.
%C A256945 The gcd of the coefficients is the same for each form within a cycle, so is a cycle invariant.  There will exist cycles with gcd invariant equal to k precisely when D(n)/k^2 = A079896(m) for some m.  In this case, the number of such cycles is A087048(m).
%D A256945 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed.  See Section 3.3 on page 359.
%H A256945 Robin Visser, <a href="/A256945/b256945.txt">Table of n, a(n) for n = 1..10000</a>
%F A256945 a(n) is the sum A087048(m) over all integers m with D(m)= D(n)/k^2  for some integer k.
%e A256945 a(6) gives the number of cycles of reduced indefinite forms of discriminant D(6) = 20.  This is the sum A087048(1) + A087048(6) = 2.
%o A256945 (SageMath)
%o A256945 def a(n):
%o A256945     i, D, S = 1, Integer(5), []
%o A256945     while(i < n):
%o A256945         D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
%o A256945     for b in range(1, isqrt(D)+1):
%o A256945         if ((D-b^2)%4 != 0): continue
%o A256945         for a in Integer((D-b^2)/4).divisors():
%o A256945             Q = BinaryQF(a, b, -(D-b^2)/(4*a))
%o A256945             if all([(not Q.is_equivalent(t)) for t in S]): S.append(Q)
%o A256945     return len(S)  # _Robin Visser_, May 31 2025
%Y A256945 Cf. A079896, A087048.
%K A256945 nonn
%O A256945 1,3
%A A256945 _Barry R. Smith_, Apr 19 2015
%E A256945 Offset corrected and more terms from _Robin Visser_, May 31 2025