cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[2, 199], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[#]] == 1 &] (* Alonso del Arte, Apr 17 2015 *)
  • PARI
    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 */
  • PARI
    is(n)=issquarefree(n) && qfbclassno(if(n%4>1, 4, 1)*n)==1 \\ Charles R Greathouse IV, Jan 19 2017

Extensions

The table in Borevich and Shafarevich extends to 497.

A029705 Squarefree n such that Q(sqrt(n)) has class number 5.

Original entry on oeis.org

401, 439, 499, 727, 817, 982, 1093, 1126, 1327, 1393, 1429, 1486, 1641, 1766, 1897, 2027, 2081, 2153, 2399, 2878, 3121, 3134, 3181, 3238, 3251, 3253, 3814, 3967, 3997, 4271, 4353, 4357, 4358, 4441, 4591, 4622, 4757, 4889, 5107, 5241, 5269, 5527, 5711, 5774
Offset: 1

Views

Author

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

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[5774], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 5 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)
  • PARI
    is(n)=issquarefree(n)&&bnfinit('x^2-n).cyc==[5] \\ Charles R Greathouse IV, 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

Views

Author

Arkadiusz Wesolowski, Oct 19 2012

Keywords

Links

Crossrefs

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

Programs

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

Views

Author

Arkadiusz Wesolowski, Oct 19 2012

Keywords

Links

Crossrefs

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

Programs

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

Views

Author

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

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

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

Keywords

Links

Crossrefs

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

Programs

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

Extensions

Initial term added by Arkadiusz Wesolowski, Oct 18 2012

A146209 Integers a(n) for which the factorization in the real quadratic field Q(sqrt(a(n))) is not unique.

Original entry on oeis.org

10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 65, 66, 70, 74, 78, 79, 82, 85, 87, 91, 95, 102, 105, 106, 110, 111, 114, 115, 119, 122, 123, 130, 138, 142, 143, 145, 146, 154, 155, 159, 165, 170, 174, 178, 182, 183, 185, 186, 187, 190, 194, 195
Offset: 1

Views

Author

Pahikkala Jussi, Oct 28 2008

Keywords

Comments

The class number of Q(sqrt(a(n))) is greater than 1.
Contains A029702, A053330 and A051990 as subsequences. See A219361 for positive integers D for which Q(sqrt D) is a UFD. - M. F. Hasler, Oct 30 2014

Examples

			For n = 6, a(6) = 35 since 35 is the sixth positive squarefree integer u for which the factorization in Q(sqrt(u)) is not unique.

References

  • Z. I. Borevich and I. R. Shafarevich, Zahlentheorie. Birkhäuser Verlag, Basel und Stuttgart (1966).

Links

Crossrefs

Cf. A003172.

Programs

  • Mathematica
    Select[Range[200], SquareFreeQ[#] && NumberFieldClassNumber[Sqrt[#]] > 1 &] (* Alonso del Arte, Sep 05 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

Views

Author

Arkadiusz Wesolowski, Oct 19 2012

Keywords

Links

Crossrefs

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

Programs

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

Views

Author

Arkadiusz Wesolowski, Oct 19 2012

Keywords

Links

Crossrefs

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

Programs

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

Views

Author

Arkadiusz Wesolowski, Oct 19 2012

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[23806], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 9 &]
  • PARI
    is(n)=issquarefree(n) && qfbclassno(if(n%4>1,4,1)*n)==9 \\ Charles R Greathouse IV, Jan 19 2017
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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