A014600 Class numbers h(D) of imaginary quadratic orders with discriminant D == 0 or 1 mod 4, D<0.

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 3, 2, 2, 2, 4, 2, 1, 3, 5, 2, 2, 2, 4, 4, 3, 2, 4, 2, 1, 4, 7, 2, 2, 3, 5, 4, 3, 4, 6, 2, 2, 3, 8, 4, 2, 2, 5, 6, 3, 3, 8, 2, 2, 6, 10, 4, 2, 3, 5, 4, 5, 4, 6, 4, 3, 6, 10, 4, 2, 2, 7, 6, 4, 4, 10, 4, 1, 8, 11, 4, 4, 3, 6, 6, 5, 4, 8, 4, 2, 5, 13, 4, 4

Offset: 0

Author

Eric Rains (rains(AT)caltech.edu)

Keywords

• nonn

Comments

The sequence consists of class numbers of imaginary quadratic "orders", not imaginary quadratic "fields". The difference is that an imaginary quadratic order may be a non-maximal order, but a class number of an imaginary quadratic field always refers to the class number of the maximal order within that imaginary quadratic field. - David Jao, Sep 13 2020

References

• H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, pp. 514-5.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
• William Cason, Akash Jim, Charlie Medlock, Erick Ross, Trevor Vilardi, and Hui Xue, Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace, arXiv:2407.11694 [math.NT], 2024. See p. 5.
• Steven R. Finch, Class number theory
• Steven R. Finch, Class number theory [Cached copy, with permission of the author]
• Erick Ross and Hui Xue, Signs of the Second Coefficients of Hecke Polynomials, arXiv:2407.10951 [math.NT], 2024. See p. 5.
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Programs

Mathematica

ClassList[n_?Negative] :=
Select[Flatten[#, 1] &@Table[
    {i, j, (j^2 - n)/(4 i)}, {i, Sqrt[-n/3]}, {j, 1 - i, i}],
  Mod[#3, 1] == 0 && #3 >= # &&
      GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &]
a[n_] := Length[ClassList[Floor[n/2]*-4 - Mod[n,2] - 3]] (* David Jao, Sep 14 2020 *)

PARI

a(n)=qfbclassno(n\2*-4-n%2-3) \\ Charles R Greathouse IV, Apr 25 2013

PARI

a(n)=quadclassunit(n\2*-4-n%2-3).no \\ Charles R Greathouse IV, Apr 25 2013

Extensions

Name corrected by David Jao, Sep 13 2020