A082174 - OEIS

%I #12 Jan 18 2021 05:10:08

%S 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,

%T 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,

%U 8,8,28,14,10,4,16,16,10,12

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

%C 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.

%D 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.

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

%e a(0)=2 because there are two reduced forms for D(0)=A079896(0)=5, namely [a,b,c]=[-1, 1, 1] and [1, 1, -1]; here f(5)=3.

%e a(4)=6: for D(4)=A079896(4)=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.

%e a(5)=2: for D(5)=A079896(5)=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].

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

%K nonn

%O 0,1

%A _Wolfdieter Lang_, Apr 11 2003