A029705 -id:A029705 - cp's OEIS frontend

A003172 Q(sqrt n) is a unique factorization domain (or simple quadratic field).

2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 22, 23, 29, 31, 33, 37, 38, 41, 43, 46, 47, 53, 57, 59, 61, 62, 67, 69, 71, 73, 77, 83, 86, 89, 93, 94, 97, 101, 103, 107, 109, 113, 118, 127, 129, 131, 133, 134, 137, 139, 141, 149, 151, 157, 158, 161, 163, 166, 167, 173, 177, 179, 181, 191, 193, 197, 199, 201

Offset: 1

N. J. A. Sloane

Squarefree numbers n such that A003649 is 1. - T. D. Noe, Apr 02 2008

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 422-423.
E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields. British Association Mathematical Tables, Vol. 4, London, 1934. (See p. 1.)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 296.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 [terms 1 through 1000 by T. D. Noe]
A. M. Odlyzko, Letters to N. J. A. Sloane Feb 1974
R. G. Underwood, On the content bound for real quadratic field extensions, Axioms 2013, 2, 1-9; doi:10.3390/axioms2010001.
Index entries for sequences related to quadratic fields

Cf. A061574 (includes negative n), A029702-A029705, A218038-A218042.

Select[Range[2, 199], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[#]] == 1 &] (* Alonso del Arte, Apr 17 2015 *)

A007947(n)={my(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]); }
{ for (n=2, 10^3,
    if ( n!=A007947(n), next() );
    K = bnfinit(x^2 - n);
    if ( K.cyc == [], print1( n, ", ") );
); }
/* Joerg Arndt, Oct 18 2012 */

is(n)=issquarefree(n) && qfbclassno(if(n%4>1, 4, 1)*n)==1 \\ Charles R Greathouse IV, Jan 19 2017

The table in Borevich and Shafarevich extends to 497.

A029702 Q(sqrt(n)) has class number 2.

Original entry on oeis.org

10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 65, 66, 70, 74, 78, 85, 87, 91, 95, 102, 105, 106, 110, 111, 114, 115, 119, 122, 123, 138, 143, 146, 154, 155, 159, 165, 174, 178, 182, 183, 185, 186, 187, 190, 194, 202, 203, 205, 215, 218, 221, 222, 230, 238, 246

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

nonn

Smallest term that is in A146209 but not this sequence is 79, since Q(sqrt(79)) has class number 3. - Alonso del Arte, Aug 25 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to quadratic fields

Cf. A003172, A029703-A029705, A218038-A218042.

Select[Range[246], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 2 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

A007947(n)={my(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]); }
{ for (n=2, 10^3,
    if ( n!=A007947(n), next() );
    K = bnfinit(x^2 - n);
    if ( K.cyc == [2], print1( n, ", ") );
); }
/* Joerg Arndt, Oct 18 2012 */

A218038 Numbers n such that Q(sqrt(n)) has class number 6.

Original entry on oeis.org

235, 346, 427, 506, 574, 697, 785, 786, 842, 874, 894, 895, 898, 899, 906, 985, 1086, 1191, 1211, 1339, 1342, 1345, 1406, 1527, 1546, 1639, 1735, 1758, 1765, 1851, 1866, 1882, 1937, 1954, 2118, 2230, 2233, 2263, 2298, 2495, 2505, 2510, 2554, 2666, 2678, 2726

Offset: 1

Arkadiusz Wesolowski, Oct 19 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Class Number

Cf. A003172, A029702-A029705, A218039-A218042.

Select[Range[2726], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 6 &]

is(n)=issquarefree(n) && qfbclassno(if(n%4>1,4,1)*n)==6 \\ Charles R Greathouse IV, Jan 19 2017

A218042 Numbers n such that Q(sqrt(n)) has class number 10.

Original entry on oeis.org

1111, 1226, 2031, 2335, 2362, 2602, 2986, 3129, 3246, 3379, 3585, 3598, 3599, 3722, 3782, 3966, 4097, 4321, 4334, 4359, 4555, 4582, 4586, 4843, 4865, 4867, 5071, 5611, 5615, 5630, 5631, 5777, 6071, 6078, 6085, 6087, 6202, 6239, 6294, 6395, 6574, 6854, 6891

Offset: 1

Arkadiusz Wesolowski, Oct 19 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Class Number

Cf. A003172, A029702-A029705, A218038-A218041.

Select[Range[6891], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 10 &]

is(n)=issquarefree(n) && qfbclassno(if(n%4>1,4,1)*n)==10 \\ Charles R Greathouse IV, Jan 19 2017

A029703 Q(sqrt(n)) has class number 3.

Original entry on oeis.org

79, 142, 223, 229, 254, 257, 321, 326, 359, 443, 469, 473, 659, 733, 761, 839, 934, 993, 1091, 1101, 1171, 1223, 1229, 1257, 1367, 1373, 1478, 1489, 1509, 1523, 1567, 1627, 1646, 1787, 1811, 1847, 1901, 1907, 1929, 1957, 1987, 2021, 2089, 2099, 2101, 2143, 2177, 2207, 2213

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

nonn

			79 is in the sequence because Z[sqrt(79)] has class number 3.
Z[sqrt(82)] has class number 4 and therefore 82 is not in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to quadratic fields

Cf. A003172, A029702, A029704-A029705, A218038-A218042.

Select[Range[2000], SquareFreeQ[#] && NumberFieldClassNumber[Sqrt[#]] == 3 &] (* Alonso del Arte, Oct 17 2012 *)

A007947(n)={my(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]); }
{ for (n=2, 10^4,
    if ( n!=A007947(n), next() );
    K = bnfinit(x^2 - n);
    if ( K.cyc == [3], print1( n, ", ") );
); }
/* Joerg Arndt, Oct 18 2012 */

Missing initial term (79) added by Alonso del Arte, Oct 17 2012

A029704 Q(sqrt(n)) has class number 4.

Original entry on oeis.org

82, 130, 145, 170, 195, 210, 219, 231, 255, 274, 290, 291, 322, 323, 330, 370, 390, 410, 434, 435, 438, 445, 455, 462, 483, 505, 510, 514, 530, 546, 555, 570, 579, 582, 595, 610, 615, 626, 627, 651, 658, 663, 674, 689, 690, 706, 714, 715, 723, 731, 754, 759, 770

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to quadratic fields

Cf. A003172, A029702-A029703, A029705, A218038-A218042.

Select[Range[770], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 4 &] (* Arkadiusz Wesolowski, Oct 18 2012 *)

is(n)=if(issquarefree(n), my(c=bnfinit('x^2-n).cyc); c==[4] || c==[2,2], 0) \\ Charles R Greathouse IV, Oct 18 2012

Initial term added by Arkadiusz Wesolowski, Oct 18 2012

A218039 Numbers n such that Q(sqrt(n)) has class number 7.

Original entry on oeis.org

577, 1009, 1087, 1294, 1601, 1761, 1934, 2029, 2251, 2302, 2467, 2913, 4139, 4229, 4702, 5039, 5273, 5417, 5743, 5827, 6151, 6598, 7919, 8097, 8311, 8462, 8661, 8773, 9029, 9049, 9101, 9289, 9326, 9539, 10117, 10313, 10357, 10713, 10957, 11021, 11053, 11269

Offset: 1

Arkadiusz Wesolowski, Oct 19 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Class Number

Cf. A003172, A029702-A029705, A218038, A218040-A218042.

Select[Range[11269], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 7 &]

is(n)=issquarefree(n) && qfbclassno(if(n%4>1,4,1)*n)==7 \\ Charles R Greathouse IV, Jan 19 2017

A218040 Numbers n such that Q(sqrt(n)) has class number 8.

Original entry on oeis.org

226, 399, 442, 646, 799, 870, 910, 994, 1023, 1122, 1155, 1239, 1290, 1299, 1351, 1443, 1446, 1590, 1705, 1743, 1785, 1914, 1947, 1995, 2010, 2019, 2035, 2134, 2210, 2211, 2310, 2379, 2402, 2410, 2415, 2434, 2451, 2482, 2490, 2605, 2705, 2706, 2730, 2739, 2751

Offset: 1

Arkadiusz Wesolowski, Oct 19 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Class Number

Cf. A003172, A029702-A029705, A218038-A218039, A218041-A218042.

Select[Range[2751], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 8 &]

is(n)=issquarefree(n) && qfbclassno(if(n%4>1,4,1)*n)==8 \\ Charles R Greathouse IV, Jan 19 2017

A218041 Numbers n such that Q(sqrt(n)) has class number 9.

Original entry on oeis.org

1129, 1654, 3137, 3719, 4409, 4534, 5521, 5623, 5878, 6809, 7573, 7873, 9998, 10273, 10721, 10814, 11027, 11641, 12323, 12409, 12657, 13069, 13691, 14159, 15374, 15629, 16321, 16382, 17273, 17989, 18633, 19441, 21023, 21781, 22497, 22502, 23003, 23806

Offset: 1

Arkadiusz Wesolowski, Oct 19 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Class Number

Cf. A003172, A029702-A029705, A218038-A218040, A218042.

Select[Range[23806], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 9 &]

is(n)=issquarefree(n) && qfbclassno(if(n%4>1,4,1)*n)==9 \\ Charles R Greathouse IV, Jan 19 2017

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

cp's OEIS Frontend

A003172 Q(sqrt n) is a unique factorization domain (or simple quadratic field).

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A029702 Q(sqrt(n)) has class number 2.

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A218038 Numbers n such that Q(sqrt(n)) has class number 6.

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A218042 Numbers n such that Q(sqrt(n)) has class number 10.

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A029703 Q(sqrt(n)) has class number 3.

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A029704 Q(sqrt(n)) has class number 4.

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A218039 Numbers n such that Q(sqrt(n)) has class number 7.

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A218040 Numbers n such that Q(sqrt(n)) has class number 8.

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