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A082175 Number of reduced indefinite quadratic forms over the integers in two variables with discriminants D(n)=A079896(n).

%I A082175 #21 Jun 01 2025 10:04:13
%S A082175 2,2,4,2,6,4,4,4,8,2,6,8,6,8,10,4,6,8,12,2,8,12,8,6,12,8,8,6,18,12,4,
%T A082175 8,16,8,12,14,8,4,16,18,6,8,20,8,14,16,14,12,6,12,16,4,14,20,16,8,20,
%U A082175 14,8,8,28,20,10,4,22,16,10,20,20,16,12,8,20,2,20,24,20,10,24,16,8,12
%N A082175 Number of reduced indefinite quadratic forms over the integers in two variables with discriminants D(n)=A079896(n).
%C A082175 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)). See the Scholz-Schoeneberg reference for this definitions.
%D A082175 A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch.IV, par.31, p. 112.
%H A082175 Robin Visser, <a href="/A082175/b082175.txt">Table of n, a(n) for n = 1..10000</a>
%H A082175 Henri Cohen, <a href="https://doi.org/10.1007/978-3-662-02945-9">A Course in Computational Algebraic Number Theory</a>, Grad. Texts in Math., Vol. 138, Springer-Verlag, Berlin, 1993. xii+534 pp.  See Definition 5.6.2 on page 257.
%F A082175 a(n)= number of reduced indefinite quadratic forms over the integers for D(n)=A079896(n) (counting also nonprimitive forms).
%e A082175 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.
%e A082175 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 (that is a,b and c are relatively prime).
%e A082175 a(6)=4: 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], Here two of them are nonprimitive, namely [-2, 2, 2], [2, 2, -2].
%e A082175 a(11)=6, D(11)=A079896(11)=32 (f(32)=6); the 6 reduced forms are [-4, 4, 1], [-2, 4, 2], [-1, 4, 4], [1, 4, -4], [2, 4, -2] and [4, 4, -1]. Two of them are nonprimitive, namely [ -2, 4, 2] and [2, 4, -2]. Therefore A082174(11)=4.
%o A082175 (SageMath)
%o A082175 def a(n):
%o A082175     i, D, ans = 1, Integer(5), 0
%o A082175     while(i < n):
%o A082175         D += 1; i += 1*(((D%4) in [0,1]) and (not D.is_square()))
%o A082175     for b in range(1, isqrt(D)+1):
%o A082175         if ((D-b^2)%4 != 0): continue
%o A082175         for a in Integer((D-b^2)/4).divisors():
%o A082175             if abs(sqrt(D) - 2*a) < b: ans += 1
%o A082175     return 2*ans  # _Robin Visser_, May 31 2025
%Y A082175 Cf. A082174 (number of primitive reduced forms).
%K A082175 nonn,easy
%O A082175 1,1
%A A082175 _Wolfdieter Lang_, Apr 11 2003
%E A082175 Offset corrected and more terms from _Robin Visser_, May 31 2025