A006374 Number of positive definite reduced binary quadratic forms of discriminant -4*n.

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

Examples

			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
		

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

Crossrefs

Cf. A006371, A006375, A096446, A096445.

Programs

Sage

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

Formula

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

Extensions

Name clarified by Wolfdieter Lang, Mar 31 2019