A006374 Number of positive definite reduced binary quadratic forms of discriminant -4*n. (Formerly M0214)

1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 4, 4, 2, 4, 4, 4, 4, 3, 4, 6, 4, 2, 6, 6, 3, 6, 6, 4, 6, 4, 6, 7, 4, 4, 8, 8, 2, 6, 8, 6, 8, 4, 4, 10, 6, 4, 10, 8, 5, 7, 8, 6, 6, 8, 8, 12, 4, 2, 12, 8, 6, 8, 10, 8, 8, 8, 4, 12, 8, 4, 14, 9, 4, 10, 10, 10, 8, 4, 10, 14, 9, 4, 12, 12, 4, 10, 12, 6, 12, 10, 8

Offset

1,3

Comments

In Hurwitz and Kritikos (HK) a definite form a*x^2 + 2*b*x*y + c*y^2, denoted by f(a,b,c), has discriminant Delta = -D = b^2 - 4*a*c < 0. Usually this is F = [a,2*b,c] with discriminant Disc = 4*(b^2 - a*c) = 4*Delta. A definite form is reduced if 2*|b| <= a <= c, and if any of the inequalities reduces to an equality then b >= 0 (HK, p. 179). The positive definite case has a > 0 and c > 0. Here the forms F do not need to satisfy gcd(a,2*b,c) = 1. - Wolfdieter Lang, Mar 31 2019

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 360.

A. Hurwitz and N. Kritikos, transl. with add. material by W . C. Schulz, "Lectures on Number Theory", Springer-Verlag, New York, 1986, p. 186.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

Formula

a(n) = A006371(2*n) for all n > 0. - Robin Visser, May 29 2025

Example

a(5) = 2 because the two forms F = [a,2*b,c] with discriminant Disc = -4*5 = -20 are [1,0,5] and [2,2,3]. ([2,-2,3] is not reduced, (-2,2,-3) is not positive definite). - Wolfdieter Lang, Mar 31 2019

Prog

(SageMath)
def a(n):
    ans = 0
    for b in range(-isqrt(n/3), isqrt(n/3)+1):
        for a in Integer(n+b^2).divisors():
            if ((2*abs(b)==a) or (a^2==n+b^2)) and (b < 0): continue
            if (a >= 2*abs(b)) and (a^2 <= n+b^2): ans += 1
    return ans  # Robin Visser, May 29 2025

Crossrefs

Cf. A006371, A006375, A000003, A096445.

Sequence in context: A229942 A025422 A078640 * A193677 A281855 A137921

Adjacent sequences: A006371 A006372 A006373 * A006375 A006376 A006377

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

Name clarified by Wolfdieter Lang, Mar 31 2019

Status

approved