A038552 -id:A038552 - cp's OEIS frontend

A038581 Absolute value of first differences of A038552, divided by 24.

11, 20, 27, 47, 45, 90, 16, 180, 134, 76, 89, 115, 396, 184, 135, 245, 471, 405, 825, 124, 1001, 220, 874, 769, 405, 15, 944, 1671, 1320, 45, 1309, 2410, 1365, 876, 280, 2446, 5460, 1519, 310, 1485, 680, 855, 795, 481, 6440, 3640, 1680, 2416, 155, 6525, 4455, 585, 4459

Offset: 1

Robert Brewer (rbrewerjr(AT)aol.com)

nonn

Eric Weisstein's World of Mathematics, Class Number

Cf. A038552.

a(n) = abs(A038552(n+1)-A038552(n))/24.

Corrected and extended by Michel Marcus, Apr 29 2016

A046125 Number of negative fundamental discriminants having class number n.

Original entry on oeis.org

9, 18, 16, 54, 25, 51, 31, 131, 34, 87, 41, 206, 37, 95, 68, 322, 45, 150, 47, 350, 85, 139, 68, 511, 95, 190, 93, 457, 83, 255, 73, 708, 101, 219, 103, 668, 85, 237, 115, 912, 109, 339, 106, 691, 154, 268, 107, 1365, 132, 345, 159, 770, 114, 427, 163, 1205, 179, 291

Offset: 1

Eric W. Weisstein

			a(1) = 9 because the discriminants {-3,-4,-7,-8,-11,-19,-43,-67,-163} are the only ones with class number 1.

Andy Huchala, Table of n, a(n) for n = 1..100 (from Watkins' paper)
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
K. Soundararajan, The number of imaginary quadratic fields with a given class number, Hardy-Ramanujan Journal, Vol. 30 (2007), pp. 13-18.
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation 73 (2004), pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields.

Cf. A014602, A014603, A006203, A013658, A046002-A046016, A048925, A056987, A081319, A038552, A191408, A191410.

Cf. A038552, A306633.

FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *);
k = 1; t = Table[0, {125}]; While[k < 2000001, If[ FundamentalDiscriminantQ@ -k, a = NumberFieldClassNumber@ Sqrt@ -k; If[a < 126, t[[a]]++]]; k++]; t (* Robert G. Wilson v Jun 01 2011 *)

lista(nn=10^7) = {my(NMAX=100, v = vector(NMAX), c); for (k=1, nn, if (isfundamental(-k), if ((c = qfbclassno(-k)) <= NMAX, v[c] ++););); v;} \\ Michel Marcus, Feb 17 2022

From Amiram Eldar, Apr 15 2025: (Start)
Formulas from Soundararajan (2007):
Sum_{k=1..n} a(k) = (3*zeta(2)/zeta(3)) * n^2 + O(n^2 * log(n)^(-1/2+eps)).
a(n) << n^2 * log(log(n))^4 / log(n). (End)

Edited by Robert G. Wilson v, May 13 2003

Corrected and extended by Dean Hickerson, May 20 2003. The values were obtained by transcribing and combining data from Tables 1-3 of Buell's paper, which has information for all values of n up to 125.

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

Original entry on oeis.org

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
```
(* First do <
				
```

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

A225060 Discriminants D < 0 such that h(D) > h(D') for D < D' < 0, negated.

Original entry on oeis.org

3, 15, 23, 39, 47, 71, 95, 119, 167, 191, 215, 239, 311, 431, 479, 551, 671, 719, 791, 839, 959, 1151, 1319, 1511, 1559, 1679, 1991, 2159, 2351, 2519, 2831, 2999, 3071, 3671, 3839, 3911, 4031, 4079, 4199, 4679, 4991, 5351, 5519, 5591, 5711, 6431, 6551, 7391, 8111

Offset: 1

Charles R Greathouse IV, Apr 26 2013

nonn

Essentially records in A014600.

H. Heilbronn, On the class number in imaginary quadratic fields, Quart. J. Math. Oxford 5 (1934), pp. 293-301.

Charles R Greathouse IV, Table of n, a(n) for n = 1..675

Cf. A014600, A225061, A038552.

r=0;forstep(n=3,1e6,[1,3],t=qfbclassno(-n);if(t>r,r=t;print1(n", ")))

A225061 Record class numbers of discriminants of imaginary quadratic fields: h(-A225060(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 19, 21, 25, 26, 30, 31, 32, 33, 36, 41, 45, 49, 51, 52, 56, 60, 63, 64, 68, 73, 76, 81, 82, 83, 84, 85, 88, 91, 92, 93, 97, 99, 109, 114, 117, 120, 121, 126, 134, 135, 138, 139, 153, 156, 161, 165, 174, 178, 181, 185, 195, 202, 205

Offset: 1

Charles R Greathouse IV, Apr 26 2013

nonn

H. Heilbronn, On the class number in imaginary quadratic fields, Quart. J. Math. Oxford 5 (1934), pp. 293-301.

Charles R Greathouse IV, Table of n, a(n) for n = 1..675

Cf. A014600, A225060, A038552.

r=0;forstep(n=3,1e6,[1,3],t=qfbclassno(-n);if(t>r,print1(r=t", ")))

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.

Original entry on oeis.org

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.

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

A038581 Absolute value of first differences of A038552, divided by 24.

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A046125 Number of negative fundamental discriminants having class number n.

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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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A225060 Discriminants D < 0 such that h(D) > h(D') for D < D' < 0, negated.

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A225061 Record class numbers of discriminants of imaginary quadratic fields: h(-A225060(n)).

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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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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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