A344072 -id:A344072 - cp's OEIS frontend

A060649 Smallest number k==3 (mod 4) such that Q(sqrt(-k)) has class number n, or 0 if no such k exists.

3, 15, 23, 39, 47, 87, 71, 95, 199, 119, 167, 231, 191, 215, 239, 399, 383, 335, 311, 455, 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

Robert G. Wilson v, Apr 17 2001

nonn

From Jianing Song, May 08 2021: (Start)
Conjecture 1: a(n) > 0 for all n;
Conjecture 2: a(n) = o(n^2). (End)
Conjecture: this is also the smallest absolute value of negative fundamental discriminant d for class number n. This is to say, for even n, if a(n) > 0 and A344072(n/2) > 0, then A344072(n/2) > a(n). - Jianing Song, Oct 03 2022

Jianing Song, Table of n, a(n) for n = 1..1162
Eric Weisstein's World of Mathematics, Class Number.

Cf. A002148, A081319, A344072, A038552.

Mathematica
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a(n) = my(d=3); while(!isfundamental(-d) || qfbclassno(-d)!=n, d+=4); d \\ Jianing Song, May 08 2021

Edited by Dean Hickerson, Mar 17 2003

Escape clause added by Jianing Song, May 08 2021

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

Dean Hickerson, Mar 18 2003

nonn

			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)

Robert G. Wilson v, Table of n, a(n) for n = 1..4500
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.

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

t[] := 0; k = -1; While[k > -3100, a = NumberFieldClassNumber@Sqrt@k; If[t[a] == 0, t[a] = -k; Print[{a, -k}]]; k--]; t /@ Range@51 (* _Robert G. Wilson v, Sep 25 2017 *)

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

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

Jianing Song, May 08 2021

nonn

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?

			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.

Jianing Song, Table of n, a(n) for n = 1..488

Cf. A060649, A344072.

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

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.

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

A357573 Largest 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

232, 1012, 1588, 3448, 5272, 8248, 9172, 14008, 21652, 21508, 26548, 32008, 45208, 53188, 57688, 65668, 73588, 85012, 121972, 120712, 117748, 137272, 189352, 162628, 174868, 201268, 194968, 249208, 188248, 332872, 341608, 424708, 370792, 411832, 377512, 539092, 332308, 486088, 369832, 435268, 604948, 667192, 548788, 601528, 596212, 566008, 737752, 795832, 645208, 802888

Offset: 1

Jianing Song, Oct 03 2022

By definition, a(n) <= 4*A038552(2n).

Conjecture: if A038552(2n) == 3 (mod 4), a(n) > 0, then a(n) < A038552(2n). If this is true, then A038552(n) is also the largest absolute value of negative fundamental discriminant d for class number n.

			a(1) = 232: h(-k) = 2 <=> k = 15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427, so the largest even k such that h(-k) = 2 is k = 232.

Cf. A038552, A344072.

cp's OEIS Frontend

A060649 Smallest number k==3 (mod 4) such that Q(sqrt(-k)) has class number n, or 0 if no such k exists.

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

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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.

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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.

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A357573 Largest 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.

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