A051990 -id:A051990 - cp's OEIS frontend

A051991 Discriminants of real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+2k, k odd.

34, 51, 66, 87, 102, 119, 123, 138, 146, 194, 215, 231, 258, 287, 318, 330, 390, 402, 410, 435, 447, 455, 482, 527, 570, 615, 623, 627, 635, 678, 770, 782, 798, 843, 890, 902, 1022, 1095, 1235, 1298, 1302, 1515, 1547, 1610, 1770, 1938, 1995

Offset: 1

N. J. A. Sloane, Jan 04 2000

Excludes discriminants appearing in A051990.

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4(2), column D.

Georg Fischer, Table of n, a(n) for n = 1..76

Cf. A051990, A051992, A051993, A051994, A051995, A051996, A051997, A051998.

A051992 Discriminants of real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+4k, k odd.

Original entry on oeis.org

85, 165, 205, 221, 285, 357, 365, 429, 533, 629, 645, 741, 957, 965, 1085, 1205, 1245, 1365, 1469, 1517, 1533, 1685, 1853, 1965, 2013, 2037, 2045, 2085, 2373, 2397, 2405, 2613, 2805, 2813, 3005, 3045, 3237, 3485, 3885, 3965, 4245, 4277

Offset: 1

N. J. A. Sloane, Jan 04 2000

Excludes discriminants appearing in A051990.

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4(3), column D.

Georg Fischer, Table of n, a(n) for n = 1..81

Cf. A051990, A051991, A051993, A051994, A051995, A051996, A051997, A051998.

A051993 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+k, k odd; sequence gives values of r.

Original entry on oeis.org

3, 4, 5, 1, 6, 2, 1, 8, 3, 2, 2, 1, 11, 12, 13, 1, 14, 2, 1, 5, 3, 16, 17, 6, 19, 4, 1, 22, 22, 23, 2, 2, 10, 2, 1, 31, 34, 34, 2, 1, 5, 7, 36, 37, 2, 1, 13, 8, 8, 1, 14, 9, 2, 46, 47, 16, 3, 4, 56, 3, 2, 58, 20, 12, 2, 3, 2, 23, 10, 2, 24, 73, 5, 4, 2, 16, 2, 4, 8, 8, 19, 98, 20, 2, 2, 4, 24, 11, 122, 4, 4, 142, 26, 68, 242, 278, 22, 32

Offset: 1

N. J. A. Sloane, Jan 04 2000

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4(1), column r.

Cf. A051990, A051991, A051992, A051994, A051995, A051996, A051997, A051998.

a(83) and following from Georg Fischer, Sep 21 2021

A051994 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+2k, k odd; sequence gives values of r.

Original entry on oeis.org

6, 7, 8, 3, 10, 11, 11, 4, 12, 14, 3, 5, 16, 17, 6, 6, 4, 20, 4, 7, 7, 3, 22, 23, 8, 5, 25, 25, 5, 26, 4, 28, 4, 29, 6, 30, 32, 11, 7, 36, 12, 13, 3, 8, 14, 44, 3, 9, 16, 7, 7, 17, 17, 4, 18, 5, 19, 4, 61, 62, 3, 2, 5, 26, 27, 17, 20, 6, 16, 39, 28, 146, 30, 23, 61, 22

Offset: 1

N. J. A. Sloane, Jan 04 2000

Excludes discriminants appearing in A051990.

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4(2), column r.

Cf. A051990, A051991, A051992, A051993, A051995, A051996, A051997, A051998.

Offset changed to 1 by Georg Fischer, Sep 21 2021

A051995 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+4k, k odd; sequence gives values of r.

Original entry on oeis.org

9, 1, 3, 1, 1, 1, 19, 7, 23, 25, 5, 9, 1, 31, 1, 7, 7, 1, 3, 1, 13, 41, 43, 3, 15, 15, 9, 3, 7, 1, 49, 17, 1, 53, 11, 11, 19, 59, 3, 1, 13, 4, 1, 23, 5, 15, 7, 11, 27, 5, 5, 1, 3, 1, 7, 1, 19, 3, 101, 7, 3, 3, 11, 25, 127, 43, 7, 27, 13, 29, 3, 31, 3, 1, 25, 59, 179, 9, 7, 47, 13

Offset: 1

N. J. A. Sloane, Jan 04 2000

Excludes discriminants appearing in A051990.

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4(3), column r.

Cf. A051990, A051991, A051992, A051993, A051994, A051996, A051997, A051998.

Offset changed to 1 by Georg Fischer, Sep 21 2021

A051996 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+k, k odd; sequence gives values of k.

Original entry on oeis.org

1, -1, 1, 5, -1, 3, -7, 1, -3, -5, 5, -11, 1, -1, 1, 13, -1, 7, -15, -3, 5, -1, 1, 3, 1, -5, 21, -1, 1, 1, -13, -15, 3, 15, -31, 1, -1, 1, 17, -35, -7, 5, -1, 1, 19, -39, -3, -5, 5, 41, 3, 5, -23, 1, 1, -3, 17, 13, -1, -19, -29, 1, -3, 5, -31, 21, 33, -3, -7, 35, 3, 1, -15, -19, -39, 5, 41, -21, -11, 11, 5, 1, 5, -55, 55, 29, 5, -11, 1, -35, 35, 1, -7, -3, 1, 1, -15, 21

Offset: 1

N. J. A. Sloane, Jan 04 2000

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4(1), column k.

Cf. A051990, A051991, A051992, A051993, A051994, A051995, A051997, A051998.

a(73) and following from Georg Fischer, Sep 21 2021

A051997 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+2k, k odd; sequence gives values of k.

Original entry on oeis.org

-1, 1, 1, 3, 1, -1, 1, -3, 1, -1, -5, 3, 1, -1, -3, 3, -5, 1, 5, -3, 3, 7, -1, -1, -3, -5, -1, 1, 5, 1, -7, -1, 7, 1, -5, 1, -1, 3, 5, 1, 3, -3, 13, 5, 3, 1, -15, -5, 3, -7, 7, -3, 3, 13, -3, -11, 3, -15, 1, -1, -21, -33, 15, 3, -3, -5, 5, -17, 7, 3, 5, 1, -5, 7, 3, -13

Offset: 1

N. J. A. Sloane, Jan 04 2000

Excludes discriminants appearing in A051990.

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4(2), column k.

Cf. A051990, A051991, A051992, A051993, A051994, A051995, A051996, A051998.

Offset changed to 1 by Georg Fischer, Sep 21 2021

A051998 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+4k, k odd; sequence gives values of k.

Original entry on oeis.org

1, -15, -5, -17, -19, -21, 1, -3, 1, 1, 5, 3, -33, 1, -35, -5, 5, -39, -13, -41, 3, 1, 1, -15, -3, 3, 5, 15, -7, -51, 1, 3, -55, 1, -5, 5, -3, 1, -21, -65, 5, 13, -69, 3, -15, 5, -11, 7, 3, -17, 17, -89, 29, -91, 13, -95, -5, 33, 1, -15, 35, 39, 11, 5, 1, 3, 19, 5, -11, 5, -51, 5, 53, -165, -7, -3, 1, 21, -29, -5, 21

Offset: 1

N. J. A. Sloane, Jan 04 2000

Excludes discriminants appearing in A051990.

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4(3), column k.

Cf. A051990, A051991, A051992, A051993, A051994, A051995, A051996, A051997.

a(69) and following from Georg Fischer, Sep 21 2021

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

Pahikkala Jussi, Oct 28 2008

nonn

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

			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.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003172.

Select[Range[200], SquareFreeQ[#] && NumberFieldClassNumber[Sqrt[#]] > 1 &] (* Alonso del Arte, Sep 05 2012 *)

A219361 Positive integers n such that the ring of integers of Q(sqrt n) is a UFD.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33, 36, 37, 38, 41, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 59, 61, 62, 63, 64, 67, 68, 69, 71, 72, 73, 75, 76, 77, 80, 81, 83, 84, 86, 88, 89, 92, 93, 94, 96, 97, 98, 99, 100

Offset: 1

Charles R Greathouse IV, Feb 19 2013

nonn

A003172 is the main entry for this sequence, which removes duplicates (i.e., for nonsquarefree n) like Q(sqrt(8)) = Q(sqrt(2)).

See A146209 for the complement (without nonsquarefree numbers like 40, ...) {10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 65, 66, 70, 74, 78, 79, ...} (supersequence of A029702, A053330 and A051990). - M. F. Hasler, Oct 30 2014

			The following are in this sequence:
  1, 4, 9, 16, ... because Z is a UFD (by the Fundamental Theorem of Arithmetic);
  2, 8, 18, 32, ... because Z[sqrt(2)] has unique factorization;
  3, 12, 27, 48, ... because Z[(1+sqrt(3))/2] has unique factorization;
  5, 20, 45, 80, ... because Z[(1+sqrt(5))/2] has unique factorization.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A029702, A061574.

Select[Range[100], NumberFieldClassNumber[Sqrt[#]] == 1 &] (* Alonso del Arte, Feb 19 2013 *)

is(n)=n=core(n); n==1 || !#bnfinit('x^2-n).cyc

cp's OEIS Frontend

A051991 Discriminants of real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+2k, k odd.

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A051992 Discriminants of real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+4k, k odd.

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A051993 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+k, k odd; sequence gives values of r.

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A051994 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+2k, k odd; sequence gives values of r.

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A051995 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+4k, k odd; sequence gives values of r.

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A051996 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+k, k odd; sequence gives values of k.

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Author

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References

Crossrefs

Extensions

A051997 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+2k, k odd; sequence gives values of k.

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Author

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Comments

References

Crossrefs

Extensions

A051998 Consider real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+4k, k odd; sequence gives values of k.

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Author

Keywords

Comments

References

Crossrefs

Extensions

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

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Programs

Mathematica

A219361 Positive integers n such that the ring of integers of Q(sqrt n) is a UFD.

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