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Smallest number k such that C(-k) is the cyclic group of order n, where C(D) is the class group of the quadratic field with discriminant D; or 0 if no such k exists.
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%I #17 May 09 2021 02:24:07

%S 3,15,23,39,47,87,71,95,199,119,167,327,191,215,239,407,383,335,311,

%T 776,431,591,647,695,479,551,983,831,887,671,719,791,839,1079,1031,

%U 959,1487,1199,1439,1271,1151,1959,1847,1391,1319,2615,3023,1751,1511,1799

%N Smallest number k such that C(-k) is the cyclic group of order n, where C(D) is the class group of the quadratic field with discriminant D; or 0 if no such k exists.

%C Different from A060649.

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

%C Conjecture 2: a(n) = o(n^2).

%C What's the next even term after a(20) = 776 and a(104) = 14024?

%H Jianing Song, <a href="/A344073/b344073.txt">Table of n, a(n) for n = 1..488</a>

%F For odd n, if a(n) > 0, then a(n) >= A060649(n). The smallest odd n such that the inequality is strict is n = 243.

%F For even n, if a(n) > 0, A060649(n) > 0 and A344072(n/2) > 0, then a(n) >= min{A060649(n), A344072(n/2)/4}. Assuming Conjecture 2 in A344072, we have a(n) >= A060649(n). The smallest n == 2 (mod 4) such that the inequality is strict is n = 342.

%e The smallest k such that c(-k) = C_12 is k = 327, so a(12) = 327.

%e The smallest k such that c(-k) = C_16 is k = 407, so a(16) = 407.

%e The smallest k such that c(-k) = C_20 is k = 776, so a(20) = 776.

%e The smallest k such that c(-k) = C_243 is k = 38231, so a(243) = 38231.

%o (PARI) a(n) = if(n==1, 3, my(d=3); while(!isfundamental(-d) || quadclassunit(-d)[2]!=[n], d++); d)

%Y Cf. A060649, A344072.

%K nonn

%O 1,1

%A _Jianing Song_, May 08 2021