A014602 -id:A014602 - cp's OEIS frontend

A133675 Negative discriminants with form class number 1 (negated).

3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, 163

Offset: 1

N. J. A. Sloane, May 16 2003

The list on p. 260 of Cox is missing -12, the list in Theorem 7.30 on p. 149 is correct. - Andrew V. Sutherland, Sep 02 2012

Let b(k) be the number of integer solutions of f(x,y) = k, where f(x,y) is the principal binary quadratic form with discriminant d<0 (i.e., f(x,y) = x^2 - (d/4)*y^2 if 4|d, x^2 + x*y + ((1-d)/4)*y^2 otherwise), then this sequence lists |d| such that {b(k)/b(1): k>=1} is multiplicative. See Crossrefs for the actual sequences. - Jianing Song, Nov 20 2019

D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989, pp. 149, 260.
D. E. Flath, Introduction to Number Theory, Wiley-Interscience, 1989.

Cf. A014602, A003173.
The sequences {b(k): k>=0}: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A033716 (d=-12), A004531 (d=-16), A028641 (d=-19), A138805 (d=-27), A033719 (d=-28), A138811 (d=-43), A318984 (d=-67), A318985 (d=-163).
The sequences {b(k)/b(1): k>=1}: A002324 (d=-3), A002654 (d=-4), A035182 (d=-7), A002325 (d=-8), A035179 (d=-11), A096936 (d=-12), A113406 (d=-16), A035171 (d=-19), A138806 (d=-27), A110399 (d=-28), A035147 (d=-43), A318982 (d=-67), A318983 (d=-163).

ok(n)={(-n)%4<2 && quadclassunit(-n).no == 1} \\ Andrew Howroyd, Jul 20 2018

Corrected by David Brink, Dec 29 2007

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

A046006 Discriminants of imaginary quadratic fields with class number 9 (negated).

Original entry on oeis.org

199, 367, 419, 491, 563, 823, 1087, 1187, 1291, 1423, 1579, 2003, 2803, 3163, 3259, 3307, 3547, 3643, 4027, 4243, 4363, 4483, 4723, 4987, 5443, 6043, 6427, 6763, 6883, 7723, 8563, 8803, 9067, 10627

Offset: 1

Eric W. Weisstein

The class group of Q[sqrt(-4027)] is isomorphic to C_3 X C_3. For all other d in this sequence, the class group of Q[sqrt(-d)] is isomorphic to C_9. - Jianing Song, Dec 01 2019

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[10700], NumberFieldClassNumber[Sqrt[-#]] == 9 &]] (* Jean-François Alcover, Jun 27 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 9};
for(n=1, 11000, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

[n for n in (1..4000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==9] # G. C. Greubel, Mar 01 2019

A046018 Discriminants of imaginary quadratic fields with class number 21 (negated).

Original entry on oeis.org

431, 503, 743, 863, 1931, 2503, 2579, 2767, 2819, 3011, 3371, 4283, 4523, 4691, 5011, 5647, 5851, 5867, 6323, 6691, 7907, 8059, 8123, 8171, 8243, 8387, 8627, 8747, 9091, 9187, 9811, 9859, 10067, 10771, 11731, 12107, 12547, 13171, 13291

Offset: 1

Eric W. Weisstein

85 discriminants in this sequence (proved).

Giovanni Resta, Table of n, a(n) for n = 1..85 (full sequence, from Weisstein's World of Mathematics)
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.

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

Reap[ For[n = 1, n < 14000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 21, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

A351680 Discriminants of imaginary quadratic fields with class number 42 (negated).

Original entry on oeis.org

1959, 2183, 2911, 3039, 3176, 3687, 3831, 4039, 4103, 4184, 4735, 4904, 4952, 5288, 5935, 5959, 6179, 6452, 6487, 6611, 6623, 6632, 6836, 7447, 7604, 7811, 7892, 7988, 8459, 8552, 8579, 8744, 8852, 9368, 9428, 9607, 10231, 10643, 10772, 10996, 11023, 11099

Offset: 1

Andy Huchala, Mar 28 2022

Sequence contains 339 terms; largest is 280267.

The class group of Q[sqrt(-d)] is isomorphic to C_42 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..339
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351679.

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] == 42]

A046016 Discriminants of imaginary quadratic fields with class number 19 (negated).

Original entry on oeis.org

311, 359, 919, 1063, 1543, 1831, 2099, 2339, 2459, 3343, 3463, 3467, 3607, 4019, 4139, 4327, 5059, 5147, 5527, 5659, 6803, 8419, 8923, 8971, 9619, 10891, 11299, 15091, 15331, 16363, 16747, 17011, 17299, 17539, 17683, 19507, 21187, 21211, 21283, 23203, 24763, 26227, 27043, 29803, 31123, 37507, 38707

Offset: 1

Eric W. Weisstein

47 discriminants in this sequence (proved).

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[ For[ n = 1, n < 40000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 19, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

A046005 Discriminants of imaginary quadratic fields with class number 8 (negated).

Original entry on oeis.org

95, 111, 164, 183, 248, 260, 264, 276, 295, 299, 308, 371, 376, 395, 420, 452, 456, 548, 552, 564, 579, 580, 583, 616, 632, 651, 660, 712, 820, 840, 852, 868, 904, 915, 939, 952, 979, 987, 995, 1032, 1043, 1060, 1092, 1128, 1131, 1155, 1195, 1204

Offset: 1

Eric W. Weisstein

131 discriminants in this sequence (almost certainly but not proved).

Andrew Howroyd, Table of n, a(n) for n = 1..131
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

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A305416.

Cf. A191410.

Union[(-NumberFieldDiscriminant[Sqrt[-#]] &) /@ Select[Range[6400], NumberFieldClassNumber[Sqrt[-#]] == 8 &]] (* Jean-François Alcover, Jun 27 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 8} \\ Andrew Howroyd, Jul 20 2018

[n for n in (1..6500) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==8] # G. C. Greubel, Mar 01 2019

A046012 Discriminants of imaginary quadratic fields with class number 15 (negated).

Original entry on oeis.org

239, 439, 751, 971, 1259, 1327, 1427, 1567, 1619, 2243, 2647, 2699, 2843, 3331, 3571, 3803, 4099, 4219, 5003, 5227, 5323, 5563, 5827, 5987, 6067, 6091, 6211, 6571, 7219, 7459, 7547, 8467, 8707, 8779, 9043, 9907, 10243, 10267, 10459, 10651

Offset: 1

Eric W. Weisstein

68 discriminants in this sequence (proved).

R. J. Mathar, Table of n, a(n) for n=1..68 (full sequence)
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[ For[n = 1, n < 12000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 15, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

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

A046008 Discriminants of imaginary quadratic fields with class number 11 (negated).

Original entry on oeis.org

167, 271, 659, 967, 1283, 1303, 1307, 1459, 1531, 1699, 2027, 2267, 2539, 2731, 2851, 2971, 3203, 3347, 3499, 3739, 3931, 4051, 5179, 5683, 6163, 6547, 7027, 7507, 7603, 7867, 8443, 9283, 9403, 9643, 9787, 10987, 13003, 13267, 14107, 14683, 15667

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.

Reap[ For[n = 1, n < 15000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 11, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 11};
for(n=1, 16000, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

[n for n in (1..16000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==11] # G. C. Greubel, Mar 01 2019

a(40)-a(41) from Giovanni Resta, Mar 20 2013

cp's OEIS Frontend

A133675 Negative discriminants with form class number 1 (negated).

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

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A046006 Discriminants of imaginary quadratic fields with class number 9 (negated).

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A046018 Discriminants of imaginary quadratic fields with class number 21 (negated).

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A351680 Discriminants of imaginary quadratic fields with class number 42 (negated).

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A046016 Discriminants of imaginary quadratic fields with class number 19 (negated).

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A046005 Discriminants of imaginary quadratic fields with class number 8 (negated).

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Sage

A046012 Discriminants of imaginary quadratic fields with class number 15 (negated).

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Mathematica

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