A344079 - OEIS

%I #13 May 09 2021 10:48:07

%S 12,16,20,72,88,92,104,128,136,164,172,184,188,192,236,243,244,256,

%T 260,264,272,276,284,292,296,316,332,336,340,342,344,348,364,372,376,

%U 388,392,396,400,416,420,440,456,468,484,488,496,504,536,548,560,576,596,600,608,612,620,637,640,644,652,664

%N Numbers m such that A060649(m) > 0 and that C(-A060649(m)) is not cyclic, where C(D) is the class group of the quadratic field with discriminant D.

%C Assume that for all n, we have A060649(n) > 0 and either A344072(n) = 0 or A344072(n) > A060649(2n), then this sequence gives the indices where A060649 and A344073 differ.

%C All terms must be nonsquarefree, since an abelian group of squarefree order must be cyclic.

%e The smallest k == 3 (mod 4) such that Q(sqrt(-k)) has class number 12 is k = -231, but the class group of Q(sqrt(-231)) is isomorphic to C_2 X C_6, which is not cyclic, so 12 is a term.

%e The smallest k == 3 (mod 4) such that Q(sqrt(-k)) has class number 20 is k = -455, but the class group of Q(sqrt(-455)) is isomorphic to C_2 X C_10, which is not cyclic, so 20 is a term.

%e The smallest k == 3 (mod 4) such that Q(sqrt(-k)) has class number 637 is k = -149519, but the class group of Q(sqrt(-149519)) is isomorphic to C_7 X C_91, which is not cyclic, so 637 is a term.

%o (PARI) isA344079(n) = my(d=3); while(!isfundamental(-d) || qfbclassno(-d)!=n, d+=4); #quadclassunit(-d)[2]>1

%Y Cf. A060649, A344073, A344072.

%K nonn

%O 1,1

%A _Jianing Song_, May 08 2021