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

1,1

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.

Links

Table of n, a(n) for n=1..62.

Example

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.

Prog

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

Crossrefs

Cf. A060649, A344073, A344072.

Sequence in context: A337700 A243538 A183052 * A240052 A363798 A070329

Adjacent sequences: A344076 A344077 A344078 * A344080 A344081 A344082

Keyword

nonn

Author

Jianing Song, May 08 2021

Status

approved