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 1-4 of 4 results.

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

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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.

A003174 Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

These integers yield norm-Euclidean real quadratic fields. There are other positive integers, e.g., D=14 or D=69, for which Q[sqrt(D)] is Euclidean, but for a Euclidean function different from the field norm.
For further references see sequence A048981 which also lists negative D corresponding to (complex) norm-Euclidean fields. - M. F. Hasler, Jan 26 2014

References

  • H. Cohn, A Second Course in Number Theory, Wiley, NY, 1962, p. 109.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 213.
  • K. Inkeri, Über den Euklidischen Algorithmus in quadratischen Zahlkörpern. Ann. Acad. Sci. Fennicae Ser. A. 1. Math.-Phys., No. 41, 1-35, 1947. [Incorrectly gives 97 as a member of this sequence.]
  • W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 2, p. 57.
  • 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. 294.

Links

Crossrefs

Cf. A003173, A003246, A048981, A187776, A263465.

Programs

Formula

a(n) = A048981(n+5). - M. F. Hasler, Jan 26 2014

Extensions

Definition corrected and comment rephrased by M. F. Hasler, Jan 26 2014
Definition corrected by Jonathan Sondow, Oct 19 2015

A003247 Complement of A003248.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers k such that A003234(k) equals the image of some x by A000201(A001950()) (see 1.20 p. 339 of Carlitz link). - Michel Marcus, Feb 02 2014
This is the function named t in [Carlitz]. - Eric M. Schmidt, Aug 14 2014

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Programs

  • PARI
    A000201(n) = floor(n*(sqrt(5)+1)/2);
A001950(n) = floor(n*(sqrt(5)+3)/2);
A003231(n) = floor(n*(sqrt(5)+5)/2);
is003234(n) = A003231(A001950(n)) == A001950(A003231(n)) - 1;
lista(nn) = {vab = vector(nn, i, A000201(A001950(i))); v003234 = select(n->is003234(n), vector(nn, i, i)); for (n=1, #v003234, if (vecsearch(vab, v003234[n]), print1(n, ", ")););} \\ Michel Marcus, Feb 02 2014

Extensions

More terms from Michel Marcus, Feb 02 2014
New definition from Eric M. Schmidt, Aug 14 2014

A236546 Discriminant of A048981(n) (= squarefree integers for which the quadratic field Q[sqrt(D)] is norm-Euclidean).

Original entry on oeis.org

-11, -7, -3, -8, -4, 8, 12, 5, 24, 28, 44, 13, 17, 76, 21, 29, 33, 37, 41, 57, 73
Offset: 1

Views

Author

M. F. Hasler, Jan 28 2014

Keywords

Comments

Note that here, values are not sorted by size, but in direct correspondence with terms of A048981. This is in contrast to A003246, where the same positive values are listed by size.

Crossrefs

Cf. A003174, A003246, A037449, A048981, A204993, A187776.

Programs

Formula

a(n) = A037449(A048981(n)) for n>5, a(n) = -A204993(-A048981(n)) for n <= 5.
Showing 1-4 of 4 results.

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

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