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 1-2 of 2 results.

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).

Crossrefs

Cf. A003172.

Programs

  • Mathematica
    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

Views

Author

Keywords

Comments

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

Examples

			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.
		

Crossrefs

Programs

  • Mathematica
    Select[Range[100], NumberFieldClassNumber[Sqrt[#]] == 1 &] (* Alonso del Arte, Feb 19 2013 *)
  • PARI
    is(n)=n=core(n); n==1 || !#bnfinit('x^2-n).cyc
Showing 1-2 of 2 results.