A096445 Number of reduced primitive positive definite binary quadratic forms of determinant n^2.

1, 1, 2, 2, 2, 4, 4, 4, 6, 4, 6, 8, 6, 8, 8, 8, 8, 12, 10, 8, 16, 12, 12, 16, 10, 12, 18, 16, 14, 16, 16, 16, 24, 16, 16, 24, 18, 20, 24, 16, 20, 32, 22, 24, 24, 24, 24, 32, 28, 20, 32, 24, 26, 36, 24, 32, 40, 28, 30, 32, 30, 32, 48, 32, 24

Offset

1,3

Comments

Equivalently, of discriminant -4n^2.

Links

Jarrod G. Sage, Table of n, a(n) for n = 1..7000

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 3rd edition, 1999, see Table 15.1.

University of Cambridge, Number Theory: Positive Definite Binary Quadratic Forms

Example

There are three reduced binary quadratic forms ax^2 + bxy +cy^2, notated as (a,b,c), with a discriminant of -36 (equivalent to determinant of 9): (1,0,9); (3,0,3); and (2,1,5). (3,0,3) is not primitive, because a, b, and c are not coprime. (1,0,9) and (2,1,5) are primitive, so there are two primitive reduced binary quadratic forms with a determinant of 9. 9 is 3^2, so a(3) = 2.

Crossrefs

Equals A000003(n^2). Cf. A006374.

Sequence in context: A187324 A334207 A323094 * A086227 A302402 A079438

Adjacent sequences: A096442 A096443 A096444 * A096446 A096447 A096448

Keyword

nonn,look

Author

N. J. A. Sloane, Aug 11 2004

Extensions

a(8) onward from Jarrod G. Sage, Jul 11 2025

Status

approved