A344073 -id:A344073 - cp's OEIS frontend

A357600 Largest 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.

163, 427, 907, 1555, 2683, 3763, 5923, 5947, 10627, 13843, 15667, 17803, 20563, 30067, 34483, 31243, 37123, 48427, 38707, 58507, 61483, 85507, 90787, 111763, 93307, 103027, 103387, 126043, 166147, 134467, 133387, 164803, 222643, 189883, 210907, 217627, 158923, 289963, 253507

Offset: 1

Jianing Song, Oct 05 2022

Different from the largest absolute value of negative fundamental discriminant d for class number n (which is equal to A038552(n) for n <= 100) at indices 8, 48, 52, 64, 68, 96, ...

Conjecture: all terms are odd.

			Let h(D) denote the class number of the quadratic field with discriminant D.
    n | Largest number k such | k' = largest number k |   C(-k')
      |    that C(-k) = C_n   |  such that h(-k) = n  |
  ----+-----------------------+-----------------------+----------
    8 |                  5947 |                  6307 |  C_2 X C_4
   48 |                333547 |                335203 | C_2 X C_24
   52 |                435163 |                439147 | C_2 X C_26
   64 |                680947 |                693067 | C_2 X C_32
   68 |                780187 |                819163 | C_2 X C_34
   96 |               1681243 |               1684027 | C_2 X C_48

Jianing Song, Table of n, a(n) for n = 1..100

Cf. A038552, A344073.

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.

Original entry on oeis.org

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

Jianing Song, May 08 2021

nonn

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.

			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.

Cf. A060649, A344073, A344072.

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

cp's OEIS Frontend

A357600 Largest 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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