A082174 Number of primitive reduced indefinite quadratic forms over the integers in two variables with discriminants D(n)=A079896(n).

2, 2, 4, 2, 6, 2, 4, 4, 8, 2, 4, 8, 6, 8, 10, 4, 4, 4, 10, 2, 8, 12, 8, 6, 12, 2, 8, 4, 18, 12, 4, 4, 12, 8, 12, 14, 8, 4, 12, 18, 6, 8, 20, 4, 14, 8, 14, 10, 4, 12, 16, 2, 8, 20, 8, 8, 20, 14, 8, 8, 28, 14, 10, 4, 16, 16, 10, 12, 20, 6, 12, 8, 20, 2, 16, 24, 12, 10, 24, 16, 8, 8, 8, 30

Offset

1,1

Comments

An indefinite quadratic form in two variables over the integers, a*x^2 + b*x*y + c*y^2 with discriminant D = b^2 - 4*a*c > 0, 0 or 1 (mod 4) and not a square, is called reduced if b>0 and f(D) - min(|2*a|,|2*c|) <= b < f(D), with f(D) := ceiling(sqrt(D)). It is called primitive if gcd(a,b,c)=1 (relative prime). See the Scholz-Schoeneberg reference for these definitions.

References

A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch.IV, par.31, p. 112 and par.27, p. 97.

Links

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

Henri Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math., Vol. 138, Springer-Verlag, Berlin, 1993. xii+534 pp. See Definition 5.6.2 on page 257.

Formula

a(n)= number of primitive reduced indefinite binary quadratic forms over the integers for D(n)=A079896(n).

Example

a(1)=2 because there are two reduced forms for D(1)=A079896(1)=5, namely [a,b,c]=[-1, 1, 1] and [1, 1, -1]; here f(5)=3.
a(5)=6: for D(5)=A079896(5)=17 (f(17)=5) the 6 reduced [a,b,c] forms are [[-2, 1, 2], [2, 1, -2], [-2, 3, 1], [-1, 3, 2], [1, 3, -2], [2, 3, -1]]. They are all primitive.
a(6)=2: for D(6)=A079896(6)=20 (f(20)=5) there are four reduced forms: [-2, 2, 2], [2, 2, -2], [-1, 4, 1] and [1, 4, -1], but only two of them are primitive, namely [-1, 4, 1] and [1, 4, -1].

Prog

(SageMath)
def a(n):
    i, D, ans = 1, Integer(5), 0
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    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 (abs(sqrt(D)-2*a)<b) and (gcd([a, b, (D-b^2)/(4*a)])==1): ans+=1
    return 2*ans  # Robin Visser, May 31 2025

Crossrefs

Cf. A082175 (number of reduced forms, nonprimitive forms included).

Sequence in context: A086087 A284476 A395619 * A278235 A074369 A323407

Adjacent sequences: A082171 A082172 A082173 * A082175 A082176 A082177

Keyword

nonn

Author

Wolfdieter Lang, Apr 11 2003

Extensions

Offset corrected and more terms from Robin Visser, May 31 2025

Status

approved