A344079 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.

12, 16, 20, 72, 88, 92, 104, 128, 136, 164, 172, 184, 188, 192, 236, 243, 244, 256, 260, 264, 272, 276, 284, 292, 296, 316, 332, 336, 340, 342, 344, 348, 364, 372, 376, 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

Offset: 1

Author

Jianing Song, May 08 2021

Keywords

• nonn

Comments

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.

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

Examples

			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.
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.
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.
		

Crossrefs

Cf. A060649, A344073, A344072.

Programs

PARI

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