A060649 Smallest squarefree number k == 3 (mod 4) such that Q(sqrt(-k)) has class number n, or 0 if no such k exists.

3, 15, 23, 39, 47, 87, 71, 95, 199, 119, 167, 231, 191, 215, 239, 399, 383, 335, 311, 455, 431, 591, 647, 695, 479, 551, 983, 831, 887, 671, 719, 791, 839, 1079, 1031, 959, 1487, 1199, 1439, 1271, 1151, 1959, 1847, 1391, 1319, 2615, 3023, 1751, 1511, 1799

Offset

1,1

Comments

From Jianing Song, May 08 2021: (Start)

Conjecture 1: a(n) > 0 for all n;

Conjecture 2: a(n) = o(n^2). (End)

Conjecture: this is also the smallest absolute value of negative fundamental discriminant d for class number n. This is to say, for even n, if a(n) > 0 and A344072(n/2) > 0, then A344072(n/2) > a(n). - Jianing Song, Oct 03 2022

Conjecture: except a(1)=3, a(n) is congruent 7 (mod 8). - Ralf Stephan, May 23 2026

Links

Jianing Song, Table of n, a(n) for n = 1..1162

Eric Weisstein's World of Mathematics, Class Number.

Mathematica

(* First do <<NumberTheory`NumberTheoryFunctions` *) a=Table[0, {50}]; Do[If[SquareFreeQ[n], c=ClassNumber[ -n]; If[c<=50&&a[[c]]==0, a[[c]]=n]], {n, 3, 3200, 4}]; a

Prog

(PARI) a(n) = my(d=3); while(!isfundamental(-d) || qfbclassno(-d)!=n, d+=4); d \\ Jianing Song, May 08 2021

Crossrefs

Cf. A002148, A081319, A344072, A038552.

Sequence in context: A106403 A225365 A225060 * A344073 A366956 A009210

Adjacent sequences: A060646 A060647 A060648 * A060650 A060651 A060652

Keyword

nonn

Author

Robert G. Wilson v, Apr 17 2001

Extensions

Edited by Dean Hickerson, Mar 17 2003

Escape clause added by Jianing Song, May 08 2021

Definition clarified by Jianing Song, Nov 04 2025

Status

approved