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A046125
Number of negative fundamental discriminants having class number n.
21
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
1,1
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.
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)
EXAMPLE
a(1) = 9 because the discriminants {-3,-4,-7,-8,-11,-19,-43,-67,-163} are the only ones with class number 1.
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 *)
PROG
(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
KEYWORD
nonn,nice
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.
STATUS
approved