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

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%I A082174 #20 Jun 01 2025 10:04:09
%S A082174 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 A082174 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 A082174 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
%N A082174 Number of primitive reduced indefinite quadratic forms over the integers in two variables with discriminants D(n)=A079896(n).
%C A082174 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 A082174 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.
%H A082174 Robin Visser, <a href="/A082174/b082174.txt">Table of n, a(n) for n = 1..10000</a>
%H A082174 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 A082174 a(n)= number of primitive reduced indefinite binary quadratic forms over the integers for D(n)=A079896(n).
%e A082174 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 A082174 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.
%e A082174 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].
%o A082174 (SageMath)
%o A082174 def a(n):
%o A082174     i, D, ans = 1, Integer(5), 0
%o A082174     while(i < n):
%o A082174         D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
%o A082174     for b in range(1, isqrt(D)+1):
%o A082174         if ((D-b^2)%4 != 0): continue
%o A082174         for a in Integer((D-b^2)/4).divisors():
%o A082174             if (abs(sqrt(D)-2*a)<b) and (gcd([a,b,(D-b^2)/(4*a)])==1): ans+=1
%o A082174     return 2*ans  # _Robin Visser_, May 31 2025
%Y A082174 Cf. A082175 (number of reduced forms, nonprimitive forms included).
%K A082174 nonn
%O A082174 1,1
%A A082174 _Wolfdieter Lang_, Apr 11 2003
%E A082174 Offset corrected and more terms from _Robin Visser_, May 31 2025