A094614 -id:A094614 - cp's OEIS frontend

A003656 Discriminants of real quadratic fields with unique factorization.

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 53, 56, 57, 61, 69, 73, 76, 77, 88, 89, 92, 93, 97, 101, 109, 113, 124, 129, 133, 137, 141, 149, 152, 157, 161, 172, 173, 177, 181, 184, 188, 193, 197, 201, 209, 213, 217, 233, 236, 237, 241, 248, 249, 253, 268, 269

Offset: 1

N. J. A. Sloane, Mira Bernstein

Discriminants of real quadratic fields with class number 1.

Other than the term 8, every term is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. In fact, for a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of the form (i), (ii) or (iii). See Theorem 1 and Theorem 2 of Ezra Brown's link. - Jianing Song, Feb 24 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 534.
H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 576.
Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003652, A003658, A014602 (imaginary case).

For discriminants of real quadratic number fields with class number 2, 3, ..., 10, see A094619, A094612-A094614, A218156-A218160; see also A035120.

maxDisc = 269; t = Table[ {NumberFieldDiscriminant[ Sqrt[n] ], NumberFieldClassNumber[ Sqrt[n] ]}, {n, Select[ Range[2, maxDisc], SquareFreeQ] } ]; Union[ Select[ t, #[[2]] == 1 && #[[1]] <= maxDisc & ][[All, 1]]] (* Jean-François Alcover, Jan 24 2012 *)

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1
A003656 = lambda n: filter(is_fund_and_qfbcn_1, (1,2,..,n))
A003656(270) # Peter Luschny, Aug 10 2014

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002

A094612 Fundamental discriminants of real quadratic number fields with class number 3.

Original entry on oeis.org

229, 257, 316, 321, 469, 473, 568, 733, 761, 892, 993, 1016, 1101, 1229, 1257, 1304, 1373, 1436, 1489, 1509, 1772, 1901, 1929, 1957, 2021, 2089, 2101, 2177, 2213, 2429, 2557, 2589, 2636, 2677, 2713, 2777, 2857, 2917, 2981, 3173, 3221, 3229, 3261, 3356, 3569

Offset: 1

Eric W. Weisstein, May 14 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094613-A094614, A218156-A218160.

Select[Range[3569], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 3 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

{ok(n) = n>10 && isfundamental(n) && qfbclassno(n)==3};
for(n=1, 3600, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 3;
A094612 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n));
A094612(3700) # G. C. Greubel, Mar 01 2019

Edited by N. J. A. Sloane, May 01 2010

A094619 Fundamental discriminants of real quadratic number fields with class number 2.

Original entry on oeis.org

40, 60, 65, 85, 104, 105, 120, 136, 140, 156, 165, 168, 185, 204, 205, 220, 221, 232, 264, 265, 273, 280, 285, 296, 305, 312, 345, 348, 357, 364, 365, 377, 380, 385, 408, 424, 429, 440, 444, 456, 460, 465, 476, 481, 485, 488, 492, 493, 533, 545, 552, 561, 565

Offset: 1

Eric W. Weisstein, May 14 2004

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094612-A094614, A218156-A218160.

Select[Range[565], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 2 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

is(n)=n>9 && isfundamental(n) && qfbclassno(n)==2 \\ Charles R Greathouse IV, Nov 05 2014

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 2;
A094619 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n));
A094619(600) # G. C. Greubel, Mar 01 2019

A218156 Fundamental discriminants of real quadratic number fields with class number 6.

Original entry on oeis.org

697, 785, 940, 985, 1345, 1384, 1708, 1765, 1937, 2024, 2233, 2296, 2505, 2941, 2993, 3021, 3144, 3281, 3305, 3368, 3496, 3576, 3580, 3592, 3596, 3624, 3973, 4065, 4344, 4764, 4765, 4844, 5073, 5353, 5356, 5368, 5369, 5529, 5621, 5624, 5685, 5901, 6108, 6153

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Andy Huchala, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218157-A218160.

Select[Range[6153], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 6 &]

A218160 Fundamental discriminants of real quadratic number fields with class number 10.

Original entry on oeis.org

3129, 3585, 4097, 4321, 4444, 4865, 4904, 5777, 6085, 6945, 7049, 7221, 7705, 8124, 8321, 8569, 9321, 9340, 9448, 9553, 9669, 10408, 10885, 11281, 11509, 11921, 11937, 11944, 12984, 12993, 12997, 13516, 13741, 13865, 14392, 14396, 14529, 14745, 14888

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218156-A218159.

Select[Range[14888], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 10 &]

A094613 Fundamental discriminants of real quadratic number fields with class number 4.

Original entry on oeis.org

145, 328, 445, 505, 520, 680, 689, 777, 780, 793, 840, 876, 897, 901, 905, 924, 1020, 1045, 1096, 1105, 1145, 1160, 1164, 1221, 1288, 1292, 1313, 1320, 1365, 1480, 1560, 1640, 1677, 1736, 1740, 1745, 1752, 1820, 1848, 1885, 1932, 2005, 2040, 2056, 2120, 2145

Offset: 1

Eric W. Weisstein, May 14 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612, A094614, A218156-A218160.

Select[Range[2145], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 4 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

{ok(n) = n>10 && isfundamental(n) && qfbclassno(n)==4};
for(n=1, 2500, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 4;
A094613 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n));
A094613(2500) # G. C. Greubel, Mar 01 2019

A081363 Smallest squarefree integer k such that Q(sqrt(k)) has class number n.

Original entry on oeis.org

2, 10, 79, 82, 401, 235, 577, 226, 1129, 1111, 1297, 730, 4759, 1534, 9871, 2305, 7054, 4954, 15409, 3601, 7057, 4762, 23593, 9634, 24859, 13321, 8761, 5626, 49281, 11665, 97753, 15130, 55339, 19882, 25601, 18226, 24337, 19834, 41614, 16899, 55966, 47959

Offset: 1

Dean Hickerson, Mar 19 2003

nonn

What is known about the asymptotics of this sequence? - Charles R Greathouse IV, Jan 26 2017

Records: 2, 10, 79, 82, 401, 577, 1129, 1297, 4759, 9871, 15409, 23593, 24859, 49281, 97753, 106537, 159199, 197137, 212137, 239119, 245023, 444089, 589822, 614849, 815413, 837929, 943951, 1025494, 1224121, 1240369, 1333255, 1334026, ..., . - Robert G. Wilson v, Apr 12 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..214

Cf. A081319, A081364, A094619, A094612, A094613, A094614.

Cf. A219361, A029702, A029703, A029704, A029705, A218038, A218039, A218040, A218041, A218042, A276541. - Robert G. Wilson v, Apr 12 2017

t[] := 0; k = 2; While[k < 56000, cn = NumberFieldClassNumber@Sqrt@k; If[ t[cn] == 0, t[cn] = k; Print[{cn, k}]]; k++]; t@# & /@ Range@ 42 (* _Robert G. Wilson v, Apr 12 2017 *)

v=vector(100); for(n=2,1e6, if(!issquarefree(n), next); t=qfbclassno(if(n%4>1,4*n,n)); if(t<=#v && v[t]==0, v[t]=n; print("a("t") = "n))) \\ Charles R Greathouse IV, Jan 26 2017

More terms from Max Alekseyev, Apr 28 2010

A218157 Fundamental discriminants of real quadratic number fields with class number 7.

Original entry on oeis.org

577, 1009, 1601, 1761, 2029, 2913, 4229, 4348, 5176, 5273, 5417, 7736, 8097, 8661, 8773, 9004, 9029, 9049, 9101, 9208, 9289, 9868, 10117, 10313, 10357, 10713, 10957, 11021, 11053, 11269, 11537, 11621, 12497, 12977, 13049, 13313, 13701, 14201, 15277, 15809

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218156, A218158-A218160.

Select[Range[15809], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 7 &]

A218158 Fundamental discriminants of real quadratic number fields with class number 8.

Original entry on oeis.org

904, 1596, 1705, 1768, 1785, 2584, 2605, 2705, 3081, 3196, 3201, 3480, 3640, 3976, 4092, 4161, 4305, 4488, 4620, 4669, 4956, 5160, 5196, 5249, 5305, 5404, 5513, 5713, 5772, 5784, 5865, 6360, 6409, 6565, 6757, 6953, 6972, 7449, 7585, 7656, 7788, 7833, 7980, 8005

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

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. A003656, A094619, A094612-A094614, A218156-A218157, A218159-A218160.

Select[Range[8005], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 8 &]

A218159 Fundamental discriminants of real quadratic number fields with class number 9.

Original entry on oeis.org

1129, 3137, 4409, 5521, 6616, 6809, 7573, 7873, 10273, 10721, 11641, 12409, 12657, 13069, 14876, 15629, 16321, 17273, 17989, 18136, 18633, 19441, 21781, 22492, 22497, 23512, 24029, 24169, 24697, 24781, 25361, 26573, 27221, 27349, 28901, 29317, 31897

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218156-A218158, A218160.

Select[Range[31897], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 9 &]

cp's OEIS Frontend

A003656 Discriminants of real quadratic fields with unique factorization.

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A094612 Fundamental discriminants of real quadratic number fields with class number 3.

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A094619 Fundamental discriminants of real quadratic number fields with class number 2.

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A218156 Fundamental discriminants of real quadratic number fields with class number 6.

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A218160 Fundamental discriminants of real quadratic number fields with class number 10.

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A094613 Fundamental discriminants of real quadratic number fields with class number 4.

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Sage

A081363 Smallest squarefree integer k such that Q(sqrt(k)) has class number n.

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Mathematica

PARI

Extensions

A218157 Fundamental discriminants of real quadratic number fields with class number 7.

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A218158 Fundamental discriminants of real quadratic number fields with class number 8.