A006371 Number of positive definite reduced binary quadratic forms of discriminant -A014601(n).

1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 2, 2, 2, 3, 3, 2, 3, 4, 2, 1, 4, 5, 4, 2, 2, 4, 4, 3, 4, 5, 4, 1, 4, 7, 3, 3, 4, 5, 6, 3, 4, 6, 2, 2, 6, 8, 6, 3, 3, 5, 6, 3, 6, 8, 4, 2, 6, 10, 4, 2, 6, 5, 7, 5, 4, 8, 4, 3, 8, 10, 8, 3, 2, 7, 6, 4, 8, 10, 6, 1, 8

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			For n = 6, the a(6) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(6) = -12 are x^2 + 3*y^2 and 2*x^2 + 2*x*y + 2*y^2.  For n = 7, the a(7) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(7) = -15 are x^2 + x*y + 4*y^2 and 2*x^2 + x*y + 2*y^2. For n = 8, the a(8) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(8) = -16 are x^2 + 4*y^2 and 2*x^2 + 2*y^2. - _Robin Visser_, May 29 2025
		

References

• H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 5th edition, 1982, p. 144.
• 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
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Crossrefs

Cf. A006374, A014601.

Programs

SageMath

def a(n):
    D, ans = 2*n+1-(n+1)%2, 0
    for b in range(-isqrt(D/3), isqrt(D/3)+1):
        if ((D+b^2)%4 != 0): continue
        for a in Integer((D+b^2)/4).divisors():
            if ((abs(b)==a) or (a^2==(D+b^2)/4)) and (b < 0): continue
            if (a >= abs(b)) and (a^2 <= (D+b^2)/4): ans += 1
    return ans  # Robin Visser, May 29 2025

Formula

a(2*n) = A006374(n) for all n > 0. - Robin Visser, May 29 2025

Extensions

More terms from Sean A. Irvine, Mar 19 2017

Name clarified and offset corrected by Robin Visser, May 29 2025