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

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, 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, 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

Offset: 1

Author

Barry R. Smith, Apr 19 2015

Keywords

• nonn

Comments

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.

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).

Examples

			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.
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed. See Section 3.3 on page 359.

Links

• Robin Visser, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A079896, A087048.

Programs

SageMath

def a(n):
    i, D, S = 1, Integer(5), []
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    for b in range(1, isqrt(D)+1):
        if ((D-b^2)%4 != 0): continue
        for a in Integer((D-b^2)/4).divisors():
            Q = BinaryQF(a, b, -(D-b^2)/(4*a))
            if all([(not Q.is_equivalent(t)) for t in S]): S.append(Q)
    return len(S)  # Robin Visser, May 31 2025

Formula

a(n) is the sum A087048(m) over all integers m with D(m)= D(n)/k^2 for some integer k.

Extensions

Offset corrected and more terms from Robin Visser, May 31 2025