A046002 -id:A046002 - cp's OEIS frontend

A046010 Discriminants of imaginary quadratic fields with class number 13 (negated).

191, 263, 607, 631, 727, 1019, 1451, 1499, 1667, 1907, 2131, 2143, 2371, 2659, 2963, 3083, 3691, 4003, 4507, 4643, 5347, 5419, 5779, 6619, 7243, 7963, 9547, 9739, 11467, 11587, 11827, 11923, 12043, 14347, 15787, 16963, 20563

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.

Reap[ For[n = 1, n < 21000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 13, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

A046014 Discriminants of imaginary quadratic fields with class number 17 (negated).

Original entry on oeis.org

383, 991, 1091, 1571, 1663, 1783, 2531, 3323, 3947, 4339, 4447, 4547, 4651, 5483, 6203, 6379, 6451, 6827, 6907, 7883, 8539, 8731, 9883, 11251, 11443, 12907, 13627, 14083, 14779, 14947, 16699, 17827, 18307, 19963, 21067, 23563, 24907, 25243, 26083, 26107, 27763, 31627, 33427, 36523, 37123

Offset: 1

Eric W. Weisstein

45 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] == 17, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

A048925 Discriminants of imaginary quadratic fields with class number 24 (negated).

Original entry on oeis.org

695, 759, 1191, 1316, 1351, 1407, 1615, 1704, 1736, 1743, 1988, 2168, 2184, 2219, 2372, 2408, 2479, 2660, 2696, 2820, 2824, 2852, 2856, 2915, 2964, 3059, 3064, 3127, 3128, 3444, 3540, 3560, 3604, 3620, 3720, 3864, 3876, 3891, 3899, 3912

Offset: 0

Eric W. Weisstein

Andy Huchala, Table of n, a(n) for n = 0..510 (first 40 terms from Eric Weisstein)
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

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

Reap[ For[n = 1, n < 4000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 24, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

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] == 24] # Andy Huchala, Feb 15 2022

A046003 Discriminants of imaginary quadratic fields with class number 6 (negated).

Original entry on oeis.org

87, 104, 116, 152, 212, 244, 247, 339, 411, 424, 436, 451, 472, 515, 628, 707, 771, 808, 835, 843, 856, 1048, 1059, 1099, 1108, 1147, 1192, 1203, 1219, 1267, 1315, 1347, 1363, 1432, 1563, 1588, 1603, 1843, 1915, 1963, 2227, 2283, 2443, 2515, 2563, 2787, 2923, 3235, 3427, 3523, 3763

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.
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938
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[3800], NumberFieldClassNumber[Sqrt[-#]] == 6 &]] (* Jean-François Alcover, Jun 27 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 6};
for(n=1, 4000, 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()==6] # G. C. Greubel, Mar 01 2019

More terms from Seiichi Manyama, Jun 03 2018

A046007 Discriminants of imaginary quadratic fields with class number 10 (negated).

Original entry on oeis.org

119, 143, 159, 296, 303, 319, 344, 415, 488, 611, 635, 664, 699, 724, 779, 788, 803, 851, 872, 916, 923, 1115, 1268, 1384, 1492, 1576, 1643, 1684, 1688, 1707, 1779, 1819, 1835, 1891, 1923, 2152, 2164, 2363, 2452, 2643, 2776, 2836, 2899, 3028

Offset: 1

Eric W. Weisstein

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

Andrew Howroyd, Table of n, a(n) for n = 1..87
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.

Cf. A191410.

Union[(-NumberFieldDiscriminant[Sqrt[-#]] &) /@ Select[Range[14000], NumberFieldClassNumber[Sqrt[-#]] == 10 &]] (* Jean-François Alcover, Jun 27 2012 *)

ok(n)={isfundamental(-n) && qfbclassno(-n) == 10} \\ Andrew Howroyd, Jul 24 2018

[n for n in (1..3500) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==10] # G. C. Greubel, Mar 01 2019

A046009 Discriminants of imaginary quadratic fields with class number 12 (negated).

Original entry on oeis.org

231, 255, 327, 356, 440, 516, 543, 655, 680, 687, 696, 728, 731, 744, 755, 804, 888, 932, 948, 964, 984, 996, 1011, 1067, 1096, 1144, 1208, 1235, 1236, 1255, 1272, 1336, 1355, 1371, 1419, 1464, 1480, 1491, 1515, 1547, 1572, 1668, 1720, 1732

Offset: 1

Eric W. Weisstein

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

Andrew Howroyd, Table of n, a(n) for n = 1..206
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.

Cf. A191410.

Reap[ For[n = 1, n < 2000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 12, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

ok(n)={isfundamental(-n) && qfbclassno(-n) == 12} \\ Andrew Howroyd, Jul 24 2018

[n for n in (1..3000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==12] # G. C. Greubel, Mar 01 2019

A046011 Discriminants of imaginary quadratic fields with class number 14 (negated).

Original entry on oeis.org

215, 287, 391, 404, 447, 511, 535, 536, 596, 692, 703, 807, 899, 1112, 1211, 1396, 1403, 1527, 1816, 1851, 1883, 2008, 2123, 2147, 2171, 2335, 2427, 2507, 2536, 2571, 2612, 2779, 2931, 2932, 3112, 3227, 3352, 3579, 3707, 3715, 3867, 3988

Offset: 1

Eric W. Weisstein

There are 95 discriminants in this sequence (almost certainly but not proved).

Andrew Howroyd, Table of n, a(n) for n = 1..95
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.

Reap[ For[n = 1, n < 4000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 14, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

ok(n)={isfundamental(-n) && qfbclassno(-n) == 14} \\ Andrew Howroyd, Jul 24 2018

A046013 Discriminants of imaginary quadratic fields with class number 16 (negated).

Original entry on oeis.org

399, 407, 471, 559, 584, 644, 663, 740, 799, 884, 895, 903, 943, 1015, 1016, 1023, 1028, 1047, 1139, 1140, 1159, 1220, 1379, 1412, 1416, 1508, 1560, 1595, 1608, 1624, 1636, 1640, 1716, 1860, 1876, 1924, 1983, 2004, 2019, 2040, 2056, 2072

Offset: 1

Eric W. Weisstein

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

Andrew Howroyd, Table of n, a(n) for n = 1..322
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.
Victor Y. Wang, On Hilbert 2-class fields and 2-towers of imaginary quadratic number fields, arXiv preprint arXiv:1508.06552, 2015
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 < 3000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 16, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

ok(n)={isfundamental(-n) && qfbclassno(-n) == 16} \\ Andrew Howroyd, Jul 24 2018

A046015 Discriminants of imaginary quadratic fields with class number 18 (negated).

Original entry on oeis.org

335, 519, 527, 679, 1135, 1172, 1207, 1383, 1448, 1687, 1691, 1927, 2047, 2051, 2167, 2228, 2291, 2315, 2344, 2644, 2747, 2859, 3035, 3107, 3543, 3544, 3651, 3688, 4072, 4299, 4307, 4568, 4819, 4883, 5224, 5315, 5464, 5492, 5539, 5899

Offset: 1

Eric W. Weisstein

The class group of Q[sqrt(-d)] is isomorphic to C_3 X C_6 for d = 9748, 12067, 16627, 17131, 19651, 22443, 23683, 34027, 34507. For all other known d in this sequence, the class group of Q[sqrt(-d)] is isomorphic to C_18. - Jianing Song, Dec 01 2019

Jianing Song, Table of n, a(n) for n = 1..150
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.

Reap[ For[n = 1, n < 6000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 18, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

A107628 Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 3, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 2, 4, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 5, 4, 0, 0, 2, 2, 0, 0, 3, 4, 0

Offset: 1

T. D. Noe, May 18 2005, Apr 30 2008

nonn

This sequence is closely related to the class number function, h(-n), which is given for fundamental discriminants in A006641. For a fundamental discriminant d, we have h(-d) < 2a(d). It appears that a(n) < Sqrt(n) for all n. For k>1, the primes p for which a(p)=k coincide with the numbers n such that the class number h(-n) is 2k-1 (see A006203, A046002, A046004, A046006. A046008, A046010, A046012, A046014, A046016 A046018, A046020). - T. D. Noe, May 07 2008

			a(15)=2 because the forms x^2 + xy + 4y^2 and 2x^2 + xy + 2y^2 have discriminant -15.

See A106856.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A106856 (start of many quadratic forms).
Cf. A133675 (n such that a(n)=1).
Cf. A223708 (without zeros).

dLim=150; cnt=Table[0, {dLim}]; nn=Ceiling[dLim/4]; Do[d=b^2-4a*c; If[GCD[a, b, c]==1 && 0<-d<=dLim, cnt[[ -d]]++ ], {b, 0, nn}, {a, b, nn}, {c, a, nn}]; cnt

{a(n)=local(m); if(n<3, 0, forvec(v=vector(3,k,[0,(n+1)\4]), if( (gcd(v)==1)&(-v[1]^2+4*v[2]*v[3]==n), m++ ), 1); m)} /* Michael Somos, May 31 2005 */

cp's OEIS Frontend

A046010 Discriminants of imaginary quadratic fields with class number 13 (negated).

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A046014 Discriminants of imaginary quadratic fields with class number 17 (negated).

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A048925 Discriminants of imaginary quadratic fields with class number 24 (negated).

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A046003 Discriminants of imaginary quadratic fields with class number 6 (negated).

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Sage

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A046007 Discriminants of imaginary quadratic fields with class number 10 (negated).

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Sage

A046009 Discriminants of imaginary quadratic fields with class number 12 (negated).

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Sage

A046011 Discriminants of imaginary quadratic fields with class number 14 (negated).

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A046013 Discriminants of imaginary quadratic fields with class number 16 (negated).

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