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

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, 21, 17, 8, 10, 14, 14, 18, 21, 13, 12, 14, 27, 11, 16, 26, 15, 17, 18, 23, 16, 10, 20, 25, 11, 13, 32, 14, 18, 26, 27, 18, 18, 38, 24, 15, 18, 28

Offset: 1

Author

Barry R. Smith, Apr 17 2015

Keywords

• nonn

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 Zagier-reduced if A>0, C>0, and B>A+C.

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).

A form is primitive if its coefficients are relatively prime.

Examples

			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.
		

References

• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981. See page 122.

Links

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

Crossrefs

Cf. A079896, A082174, A257003.

Programs

Mathematica

Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
        Mod[# - n, 2] == 0 &]}, {a,
       Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
     UnsameQ[#, {}] &], 1],
   GCD[#[[1]], #[[2]] +
       2*#[[1]], #[[1]] + #[[2]] - (n - #[[2]]^2)/(4*#[[1]])] == 1 &]], {n,
  Select[Range[
    153], ! IntegerQ[Sqrt[#]] && (Mod[#, 4] == 0 || Mod[#, 4] == 1) &]}]

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 k in range(-isqrt(D), isqrt(D)+1):
        if ((D-k^2)%4 != 0): continue
        for h in Integer((D-k^2)/4).divisors():
            if gcd([h, k+2*h, (k+h-(D-k^2)/(4*h))])==1:
                if h > (sqrt(D)-k)/2: ans += 1
    return ans  # Robin Visser, Jun 01 2025

Extensions

Offset corrected by Robin Visser, Jun 01 2025