cp's OEIS Frontend

A257004 Number of primitive Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).

%I A257004 #15 Jun 07 2025 08:14:47
%S A257004 1,2,3,3,5,4,4,6,7,5,5,10,7,10,11,9,6,8,10,7,10,16,12,11,16,8,10,12,
%T A257004 21,17,8,10,14,14,18,21,13,12,14,27,11,16,26,15,17,18,23,16,10,20,25,
%U A257004 11,13,32,14,18,26,27,18,18,38,24,15,18,28
%N A257004 Number of primitive Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).
%C A257004 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 Zagier-reduced if A>0, C>0, and B>A+C.
%C A257004 This definition is from Zagier's 1981 book, and differs from the older and more common notion of reduced form due to Lagrange (see A082175 for this definition).
%C A257004 A form is primitive if its coefficients are relatively prime.
%D A257004 D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.  See page 122.
%H A257004 Robin Visser, <a href="/A257004/b257004.txt">Table of n, a(n) for n = 1..10000</a>
%e A257004 For D=20, the a(6)=4 Zagier-reduced primitive forms are x^2+6*x*y+4*y^2, 4*x^2+6*x*y+y^2, 4*x^2+10*x*y+5*y^2, and 5*x^2+10*x*y+4*y^2.
%t A257004 Table[Length[
%t A257004   Select[Flatten[
%t A257004     Select[
%t A257004      Table[{a, k}, {k,
%t A257004        Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
%t A257004         Mod[# - n, 2] == 0 &]}, {a,
%t A257004        Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
%t A257004      UnsameQ[#, {}] &], 1],
%t A257004    GCD[#[[1]], #[[2]] +
%t A257004        2*#[[1]], #[[1]] + #[[2]] - (n - #[[2]]^2)/(4*#[[1]])] == 1 &]], {n,
%t A257004   Select[Range[
%t A257004     153], ! IntegerQ[Sqrt[#]] && (Mod[#, 4] == 0 || Mod[#, 4] == 1) &]}]
%o A257004 (SageMath)
%o A257004 def a(n):
%o A257004     i, D, ans = 1, Integer(5), 0
%o A257004     while(i < n):
%o A257004         D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
%o A257004     for k in range(-isqrt(D), isqrt(D)+1):
%o A257004         if ((D-k^2)%4 != 0): continue
%o A257004         for h in Integer((D-k^2)/4).divisors():
%o A257004             if gcd([h, k+2*h, (k+h-(D-k^2)/(4*h))])==1:
%o A257004                 if h > (sqrt(D)-k)/2: ans += 1
%o A257004     return ans  # _Robin Visser_, Jun 01 2025
%Y A257004 Cf. A079896, A082174, A257003.
%K A257004 nonn
%O A257004 1,2
%A A257004 _Barry R. Smith_, Apr 17 2015
%E A257004 Offset corrected by _Robin Visser_, Jun 01 2025