A006203 -id:A006203 - cp's OEIS frontend

A046002 Discriminants of imaginary quadratic fields with class number 5 (negated).

47, 79, 103, 127, 131, 179, 227, 347, 443, 523, 571, 619, 683, 691, 739, 787, 947, 1051, 1123, 1723, 1747, 1867, 2203, 2347, 2683

Offset: 1

Eric W. Weisstein

Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998) 295-330.
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046003-A046020.

Cf. A191410.

Union[(-NumberFieldDiscriminant[Sqrt[-#]] &) /@ Select[Range[2700], NumberFieldClassNumber[Sqrt[-#]] == 5 &]] (* Jean-François Alcover, Jun 27 2012 *)

select(n->qfbclassno(-n)==5,vector(670,n,4*n+3)) \\ Charles R Greathouse IV, Apr 25 2013

[n for n in (1..3000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==5] # G. C. Greubel, Mar 01 2019

A014603 Discriminants of imaginary quadratic fields with class number 2 (negated).

Original entry on oeis.org

15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427

Offset: 1

Eric Rains (rains(AT)caltech.edu)

Includes only fundamental discriminants. The list of non-fundamental imaginary quadratic discriminants with class number 2 (negated) is 32, 36, 48, 60, 64, 72, 75, 99, 100, 112, 147. - Andrew V. Sutherland, Apr 08 2010

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 229.

A. Abatzoglou, A. Silverberg, A. V. Sutherland, A, Wong, A framework for deterministic primality proving using elliptic curves with complex multiplication, arXiv:1404.0107 [math.NT], 2014.
Alexandre Gélin, Everett W. Howe, and Christophe Ritzenthaler, Principally Polarized Squares of Elliptic Curves with Field of Moduli Equal To Q, arXiv:1806.03826 [math.NT], 2018 (see table 1 page 4).
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A046002, ..., A046020.

Cf. A191410.

Union[ (-NumberFieldDiscriminant[ Sqrt[-#]] &) /@ Select[ Range[500], NumberFieldClassNumber[ Sqrt[-#]] == 2 &]] (* Jean-François Alcover, Jan 04 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 2} \\ Andrew Howroyd, Jul 20 2018

[n for n in (1..500) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==2] # G. C. Greubel, Mar 01 2019

Offset corrected by Jianing Song, Aug 29 2018

A046020 Discriminants of imaginary quadratic fields with class number 23 (negated).

Original entry on oeis.org

647, 1039, 1103, 1279, 1447, 1471, 1811, 1979, 2411, 2671, 3491, 3539, 3847, 3923, 4211, 4783, 5387, 5507, 5531, 6563, 6659, 6703, 7043, 9587, 9931, 10867, 10883, 12203, 12739, 13099, 13187, 15307, 15451, 16267, 17203, 17851, 18379, 20323

Offset: 1

Eric W. Weisstein

Giovanni Resta, Table of n, a(n) for n = 1..68 (full sequence, from Steven Arno et al.)
Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998) 295-330.
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Reap[ Do[ If[ NumberFieldClassNumber[ Sqrt[-n] ] == 23, d = -NumberFieldDiscriminant[ Sqrt[-n] ]; Print[d]; Sow[d]], {n, 1, 21000}]][[2, 1]] // Union (* Jean-François Alcover, Oct 22 2012 *)

select(n->qfbclassno(-n)==23, vector(22696, n, 4*n+3)) \\ Charles R Greathouse IV, Apr 25 2013

68 discriminants in this sequence (proved).

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.

A056987 Discriminants of imaginary quadratic fields with class number 25 (negated).

Original entry on oeis.org

479, 599, 1367, 2887, 3851, 4787, 5023, 5503, 5843, 7187, 7283, 7307, 7411, 8011, 8179, 9227, 9923, 10099, 11059, 11131, 11243, 11867, 12211, 12379, 12451, 12979, 14011, 14923, 15619, 17483, 18211, 19267, 19699, 19891, 20347, 21107, 21323

Offset: 1

Eric W. Weisstein

Sequence contains 95 members; largest is 93307.

The class group of Q[sqrt(-d)] is isomorphic to C_5 X C_5 for d = 12451 and 37363. For all other d in this sequence, the class group of Q[sqrt(-d)] is isomorphic to C_25. - Jianing Song, Dec 01 2019

Jianing Song, Table of n, a(n) for n = 1..95
Eric Weisstein's World of Mathematics, Class Number.

Cf. A014602, A014603, A006203, A013658, A046002, A046003, A046004, A046005, A046006, A046007, A046008, A046009, A046010, A046011, A046012, A046013, A046014, A046015, A046016, A123563, A046018, A171724, A046020, A048925.

Reap[ For[n = 1, n < 22000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 25, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

A351664 Discriminants of imaginary quadratic fields with class number 26 (negated).

Original entry on oeis.org

551, 951, 1247, 1256, 1735, 1832, 2651, 2771, 2792, 2823, 2839, 2984, 3092, 3327, 3368, 3611, 3736, 3903, 3992, 4052, 4207, 4244, 4376, 4739, 5123, 5435, 5524, 5891, 6059, 6443, 6515, 6587, 6676, 6847, 6891, 6964, 7156, 8003, 8339, 8383, 8408, 8691, 8743

Offset: 1

Andy Huchala, Feb 16 2022

Sequence contains 190 terms; largest is 103027.

The class group of Q[sqrt(-d)] is isomorphic to C_26 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..190
Eric Weisstein's World of Mathematics, Class Number.

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987.

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 26]

A002148 Smallest prime p==3 (mod 8) such that Q(sqrt(-p)) has class number 2n+1.

Original entry on oeis.org

3, 59, 131, 251, 419, 659, 1019, 971, 1091, 2099, 1931, 1811, 3851, 3299, 2939, 3251, 4091, 4259, 8147, 5099, 9467, 6299, 6971, 8291, 8819, 14771, 22619, 9539, 13331, 18443, 11171, 16979, 12011, 13859, 16931, 17939, 28211, 19211, 24251, 20411

Offset: 0

N. J. A. Sloane and Mira Bernstein

nonn

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Broadhurst and T. D. Noe, Table of n, a(n) for n = 0..10399
D. Shanks, Review of R. B. Lakein and S. Kuroda, Tables of class numbers h(-p) for fields Q(sqrt(-p)), p <= 465071, Math. Comp., 24 (1970), 491-492.

Cf. A002143 (class numbers), A002149, A003173, A006203.

a=Table[0, {101}]; Do[If[PrimeQ[m], c=NumberFieldClassNumber[Sqrt[-m]]; If[c<102 && a[[c]]==0, a[[c]]=m]], {m, 3, 30000, 8}]; Table[a[[n]], {n, 1, 101, 2}]

More terms from Robert G. Wilson v, Apr 17 2001

Edited by Dean Hickerson, Mar 17 2003

A191410 Class number, k, of n, i.e.; imaginary quadratic fields negated Q(sqrt(-n))=k, or 0 if n is not a fundamental discriminant (A003657).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 4, 4, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 3, 4, 0, 0, 6, 2, 0, 0, 2, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 5, 6, 0

Offset: 1

Robert G. Wilson v, Jun 01 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Steven Arno, M. L. Robinson and Ferrel S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arithm. 83.4 (1998), 295-330
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

a(n)= 0: n/a The complement of A003657; a(n)= 1: A014602; a(n)= 2: A014603; a(n)= 3: A006203; a(n)= 4: A013658; a(n)= 5: A046002; a(n)= 6: A046003; a(n)= 7: A046004; a(n)= 8: A046005; a(n)= 9: A046006; a(n)=10: A046007; a(n)=11: A046008; a(n)=12: A046009; a(n)=13: A046010; a(n)=14: A046011; a(n)=15: A046012; a(n)=16: A046013; a(n)=17: A046014; a(n)=18: A046015; a(n)=19: A046016; a(n)=20: A123563; a(n)=21: A046018; a(n)=22: A171724; a(n)=23: A046020; a(n)=24: A048925; a(n)=25: A056987; etc.

Cf. A003657, A191408.

FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *);
f[n_] := If[ !FundamentalDiscriminantQ@ -n, 0, NumberFieldClassNumber@ Sqrt@ -n]; Array[f, 105]

a(n)=if(isfundamental(-n),qfbclassno(-n),0) \\ Charles R Greathouse IV, Nov 20 2012

A005847 Imaginary quadratic fields with class number 2 (a finite sequence).

Original entry on oeis.org

5, 6, 10, 13, 15, 22, 35, 37, 51, 58, 91, 115, 123, 187, 235, 267, 403, 427

Offset: 1

N. J. A. Sloane

n such that Q(sqrt(-n)) has class number 2.

The PARI code lists the imaginary quadratic fields Q(sqrt(-d)) with small class number and produces A003173 (class number 1), A005847 (2), A006203 (3).

J. M. Masley, Where are the number fields with small class number?, pp. 221-242 of "Number Theory, Carbondale 1979", Lect. Notes Math. 751 (1982).
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 142.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 143-144.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Steven Arno, M. L. Robinson, and Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998), pp. 295-330.
David Masser, Alan Baker, arXiv:2010.10256 [math.HO], 2020. See p. 24.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers.
Index entries for sequences related to quadratic fields.

Cf. A003173, A005847, A006203.

Select[Range[200], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[-#]] == 2 &] (* Alonso del Arte, May 28 2015 *)

{ bnd = 10000; S = vector(10,X,[]); for (i = 1, bnd, if (issquarefree(i), n = qfbclassno(if(i%4==3,-i,-4*i)); if (n<11, S[n] = concat(S[n],i), ), )); } \\ Robert Harley (Robert.Harley(AT)inria.fr)

A046004 Discriminants of imaginary quadratic fields with class number 7 (negated).

Original entry on oeis.org

71, 151, 223, 251, 463, 467, 487, 587, 811, 827, 859, 1163, 1171, 1483, 1523, 1627, 1787, 1987, 2011, 2083, 2179, 2251, 2467, 2707, 3019, 3067, 3187, 3907, 4603, 5107, 5923

Offset: 1

Eric W. Weisstein

Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998) 295-330.
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Cf. A191410.

Union[(-NumberFieldDiscriminant[Sqrt[-#]] &) /@ Select[Range[6000], NumberFieldClassNumber[Sqrt[-#]] == 7 &]] (* Jean-François Alcover, Jun 27 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 7};
for(n=1, 6000, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

[n for n in (1..6000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==7] # G. C. Greubel, Mar 01 2019

cp's OEIS Frontend

A046002 Discriminants of imaginary quadratic fields with class number 5 (negated).

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A014603 Discriminants of imaginary quadratic fields with class number 2 (negated).

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A046020 Discriminants of imaginary quadratic fields with class number 23 (negated).

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A046125 Number of negative fundamental discriminants having class number n.

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A056987 Discriminants of imaginary quadratic fields with class number 25 (negated).

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A351664 Discriminants of imaginary quadratic fields with class number 26 (negated).

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Sage

A002148 Smallest prime p==3 (mod 8) such that Q(sqrt(-p)) has class number 2n+1.

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A191410 Class number, k, of n, i.e.; imaginary quadratic fields negated Q(sqrt(-n))=k, or 0 if n is not a fundamental discriminant (A003657).

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