A046125 Number of negative fundamental discriminants having class number n.

9, 18, 16, 54, 25, 51, 31, 131, 34, 87, 41, 206, 37, 95, 68, 322, 45, 150, 47, 350, 85, 139, 68, 511, 95, 190, 93, 457, 83, 255, 73, 708, 101, 219, 103, 668, 85, 237, 115, 912, 109, 339, 106, 691, 154, 268, 107, 1365, 132, 345, 159, 770, 114, 427, 163, 1205, 179, 291

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• nice

Examples

			a(1) = 9 because the discriminants {-3,-4,-7,-8,-11,-19,-43,-67,-163} are the only ones with class number 1.
		

Links

• Andy Huchala, Table of n, a(n) for n = 1..100 (from Watkins' paper)
• Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
• K. Soundararajan, The number of imaginary quadratic fields with a given class number, Hardy-Ramanujan Journal, Vol. 30 (2007), pp. 13-18.
• Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation 73 (2004), pp. 907-938.
• Eric Weisstein's World of Mathematics, Class Number.
• Index entries for sequences related to quadratic fields.

Crossrefs

Cf. A014602, A014603, A006203, A013658, A046002-A046016, A048925, A056987, A081319, A038552, A191408, A191410.

Cf. A038552, A306633.

Programs

Mathematica

FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *);
k = 1; t = Table[0, {125}]; While[k < 2000001, If[ FundamentalDiscriminantQ@ -k, a = NumberFieldClassNumber@ Sqrt@ -k; If[a < 126, t[[a]]++]]; k++]; t (* Robert G. Wilson v Jun 01 2011 *)

PARI

lista(nn=10^7) = {my(NMAX=100, v = vector(NMAX), c); for (k=1, nn, if (isfundamental(-k), if ((c = qfbclassno(-k)) <= NMAX, v[c] ++););); v;} \\ Michel Marcus, Feb 17 2022

Formula

From Amiram Eldar, Apr 15 2025: (Start)

Formulas from Soundararajan (2007):

Sum_{k=1..n} a(k) = (3*zeta(2)/zeta(3)) * n^2 + O(n^2 * log(n)^(-1/2+eps)).

a(n) << n^2 * log(log(n))^4 / log(n). (End)

Extensions

Edited by Robert G. Wilson v, May 13 2003

Corrected and extended by Dean Hickerson, May 20 2003. The values were obtained by transcribing and combining data from Tables 1-3 of Buell's paper, which has information for all values of n up to 125.