A307359 Class number a(n) of indefinite binary quadratic forms with discriminant 4*A000037(n) for n >= 1.

1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 4, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 2, 2, 2, 4, 4, 3, 2, 4, 4, 1, 4, 2, 2, 2, 2, 2, 4, 2, 4, 2, 1, 2, 4, 4, 2, 2, 2, 4, 1, 2, 4, 2, 4, 2, 2, 2, 4, 2, 4, 1, 2, 4, 2, 2, 4, 6, 4, 4, 2, 4, 2, 2, 4, 4, 1, 4, 4, 2, 2, 2, 4, 4, 1, 2, 8, 3, 4, 2, 4, 4, 2

Offset: 1

Author

Wolfdieter Lang, Apr 04 2019

Keywords

• nonn

Comments

This is a subsequence of A087048, See the formula.

This sequence is relevant for the Pell forms [1, 0, - D(n)], with D(n) = A000037(n) and discriminant 4*D(n).

The Buell reference, Table 2B, pp. 241-243, gives only the class numbers, called there H, for A000037(n) squarefree and not congruent to 1 modulo 4. E.g., a(3), related to discriminant 4*5 = 20, is not treated there; also a(6) for discriminant 32 = 4*(2*2^2) does not appear there.

For the a(n) cycles of primitive reduced forms of discriminant 4*A000037(n) see the W. lang link in A324251, Table 2 and Table 1, for n = 1..30. - Wolfdieter Lang, Apr 19 2019

Examples

			a(1) = 1 because 4*A000037(1) = 4*2 = 8 = A079896(e(1)) with e(1) = 1 and A087048(1) = 1.
a(12) = 4 because the twelfth even number of A079896 is 60 at position e(12) = 22, and A087048(22) = 4.
The cycle for discriminant 8 is [[1, 2, -1], [-1, 2, 1]].
The four 2-cycles for discriminant 60 are  [[1, 6, -6], [-6, 6, 1]], [[-1, 6, 6], [6, 6, -1]], [[2, 6, -3], [-3, 6, 2]] and  [[-2, 6, 3], [3, 6, -2]].
		

References

• D. A. Buell, Binary Quadratic Forms, Springer, 1989.

Links

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

Crossrefs

Cf. A000037, A079896, A082174, A087048.

Programs

SageMath

def a(n):
    i, D, S = 1, 4*n + 4*floor(1/2 + sqrt(n)), []
    for b in range(1, isqrt(D)+1):
        if ((D-b^2)%4 != 0): continue
        for a in Integer((D-b^2)/4).divisors():
            if gcd([a, b, (D-b^2)/(4*a)]) > 1: continue
            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, Jun 01 2025

Formula

a(n) gives the number of distinct cycles of primitive reduced forms of discriminant 4*A000037(n).

a(n) = A087048(e(n)), with e(n) the position of the n-th even term of A079896, for n >= 1.

Extensions

a(40) corrected and more terms from Robin Visser, Jun 01 2025