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

%I A014600 #43 Jul 25 2024 11:06:18
%S A014600 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,
%T A014600 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,
%U A014600 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
%N A014600 Class numbers h(D) of imaginary quadratic orders with discriminant D == 0 or 1 mod 4, D<0.
%C A014600 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
%D A014600 H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, pp. 514-5.
%H A014600 Charles R Greathouse IV, <a href="/A014600/b014600.txt">Table of n, a(n) for n = 0..10000</a>
%H A014600 William Cason, Akash Jim, Charlie Medlock, Erick Ross, Trevor Vilardi, and Hui Xue, <a href="https://arxiv.org/abs/2407.11694">Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace</a>, arXiv:2407.11694 [math.NT], 2024. See p. 5.
%H A014600 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Class number theory</a>
%H A014600 Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]
%H A014600 Erick Ross and Hui Xue, <a href="https://arxiv.org/abs/2407.10951">Signs of the Second Coefficients of Hecke Polynomials</a>, arXiv:2407.10951 [math.NT], 2024. See p. 5.
%H A014600 Rick L. Shepherd, <a href="http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
%t A014600 ClassList[n_?Negative] :=
%t A014600 Select[Flatten[#, 1] &@Table[
%t A014600     {i, j, (j^2 - n)/(4 i)}, {i, Sqrt[-n/3]}, {j, 1 - i, i}],
%t A014600   Mod[#3, 1] == 0 && #3 >= # &&
%t A014600       GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &]
%t A014600 a[n_] := Length[ClassList[Floor[n/2]*-4 - Mod[n,2] - 3]] (* _David Jao_, Sep 14 2020 *)
%o A014600 (PARI) a(n)=qfbclassno(n\2*-4-n%2-3) \\ _Charles R Greathouse IV_, Apr 25 2013
%o A014600 (PARI) a(n)=quadclassunit(n\2*-4-n%2-3).no \\ _Charles R Greathouse IV_, Apr 25 2013
%K A014600 nonn
%O A014600 0,7
%A A014600 Eric Rains (rains(AT)caltech.edu)
%E A014600 Name corrected by _David Jao_, Sep 13 2020