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

1,3

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).

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

Formula

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

Example

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.

Prog

(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

Crossrefs

Cf. A079896, A087048.

Sequence in context: A183015 A183018 A067390 * A282091 A354110 A015718

Adjacent sequences: A256942 A256943 A256944 * A256946 A256947 A256948

Keyword

nonn

Author

Barry R. Smith, Apr 19 2015

Extensions

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

Status

approved