cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Jianing Song, May 08 2021

Keywords

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

A344072 Smallest even k such that h(-k) = 2n, where h(D) is the class number of the quadratic field with discriminant D; or 0 if no such k exists.

Original entry on oeis.org

20, 56, 104, 164, 296, 356, 404, 584, 1172, 776, 1076, 1316, 1256, 1364, 1844, 1784, 2456, 2504, 4916, 2756, 3176, 3416, 3764, 4424, 4436, 5924, 6296, 4616, 5144, 5444, 10484, 6536, 9236, 7124, 7796, 7556, 12776, 9176, 8564, 10856, 11156, 10436, 11864, 12536, 14804, 13604, 13844, 16376, 15896, 13796
Offset: 1

Views

Author

Jianing Song, May 08 2021

Keywords

Comments

In other words, a(n) is the smallest even k such that Q(sqrt(-k/4)) has class number n; or 0 if no such k exists.
Conjecture 1: a(n) > 0 for all n.
Conjecture 2: If a(n) > 0 and A060649(2n) > 0, then we have a(n) > A060649(2n). This would imply that all terms in A225060 are odd.
Conjecture 3: There exists a positive constant c such that a(n) < c*A060649(2n) for all n.
It seems that the ratio a(n)/A060649(2n) reaches its minimum at n = 3. Among the first 250 terms, the maximum of a(n)/A060649(2n) is ~5.3116, which is attained at n = 227.

Examples

			The smallest even k such that h(-k) = 2 is k = 20, so a(1) = 20.
The smallest even k such that h(-k) = 4 is k = 56, so a(2) = 56.
The smallest even k such that h(-k) = 12 is k = 356, so a(6) = 356.

Links

Crossrefs

Cf. A060649, A225060.

Programs

  • PARI
    a(n) = my(d=4); while(!isfundamental(-d) || qfbclassno(-d)!=2*n, d+=4); d

A081319 Smallest squarefree integer k such that Q(sqrt(-k)) has class number n, or 0 if no such k exists.

Original entry on oeis.org

1, 5, 23, 14, 47, 26, 71, 41, 199, 74, 167, 89, 191, 101, 239, 146, 383, 293, 311, 194, 431, 269, 647, 329, 479, 314, 983, 341, 887, 461, 719, 446, 839, 614, 1031, 626, 1487, 1199, 1439, 689, 1151, 794, 1847, 854, 1319, 941, 3023, 1106, 1511, 1109, 1559
Offset: 1

Views

Author

Dean Hickerson, Mar 18 2003

Keywords

Examples

			From _Jianing Song_, May 08 2021: (Start)
a(6) = min{A060649(6), A344072(3)/4} = min{87, 104/4} = 26.
a(12) = min{A060649(12), A344072(6)/4} = min{231, 356/4} = 89.
a(18) = min{A060649(12), A344072(9)/4} = min{335, 1172/4} = 293.
a(38) = min{A060649(38), A344072(19)/4} = min{1199, 4916/4} = 1199. (End)

Links

Crossrefs

Cf. A060649, A038552, A046125, A014602-A046020.

Programs

Formula

a(n) = A060649(n) for odd n > 1. For even n, assuming that A060649(n) > 0 and A344072(n/2) > 0, a(n) = min{A060649(n), A344072(n/2)/4}. - Jianing Song, May 08 2021

Extensions

Edited by Max Alekseyev, Apr 28 2010
Escape clause added by Jianing Song, May 08 2021

A344073 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.

Original entry on oeis.org

3, 15, 23, 39, 47, 87, 71, 95, 199, 119, 167, 327, 191, 215, 239, 407, 383, 335, 311, 776, 431, 591, 647, 695, 479, 551, 983, 831, 887, 671, 719, 791, 839, 1079, 1031, 959, 1487, 1199, 1439, 1271, 1151, 1959, 1847, 1391, 1319, 2615, 3023, 1751, 1511, 1799
Offset: 1

Views

Author

Jianing Song, May 08 2021

Keywords

Comments

Different from A060649.
Conjecture 1: a(n) > 0 for all n;
Conjecture 2: a(n) = o(n^2).
What's the next even term after a(20) = 776 and a(104) = 14024?

Examples

			The smallest k such that c(-k) = C_12 is k = 327, so a(12) = 327.
The smallest k such that c(-k) = C_16 is k = 407, so a(16) = 407.
The smallest k such that c(-k) = C_20 is k = 776, so a(20) = 776.
The smallest k such that c(-k) = C_243 is k = 38231, so a(243) = 38231.

Links

Crossrefs

Cf. A060649, A344072.

Programs

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

Formula

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.
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.
Showing 1-4 of 4 results.

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