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

%I A060649 #31 Feb 16 2025 08:32:44
%S A060649 3,15,23,39,47,87,71,95,199,119,167,231,191,215,239,399,383,335,311,
%T A060649 455,431,591,647,695,479,551,983,831,887,671,719,791,839,1079,1031,
%U A060649 959,1487,1199,1439,1271,1151,1959,1847,1391,1319,2615,3023,1751,1511,1799
%N A060649 Smallest number k==3 (mod 4) such that Q(sqrt(-k)) has class number n, or 0 if no such k exists.
%C A060649 From _Jianing Song_, May 08 2021: (Start)
%C A060649 Conjecture 1: a(n) > 0 for all n;
%C A060649 Conjecture 2: a(n) = o(n^2). (End)
%C A060649 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
%H A060649 Jianing Song, <a href="/A060649/b060649.txt">Table of n, a(n) for n = 1..1162</a>
%H A060649 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ClassNumber.html">Class Number</a>.
%t A060649 (* 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
%o A060649 (PARI) a(n) = my(d=3); while(!isfundamental(-d) || qfbclassno(-d)!=n, d+=4); d \\ _Jianing Song_, May 08 2021
%Y A060649 Cf. A002148, A081319, A344072, A038552.
%K A060649 nonn
%O A060649 1,1
%A A060649 _Robert G. Wilson v_, Apr 17 2001
%E A060649 Edited by _Dean Hickerson_, Mar 17 2003
%E A060649 Escape clause added by _Jianing Song_, May 08 2021