A003172 Q(sqrt n) is a unique factorization domain (or simple quadratic field). (Formerly M0618)

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

1,1

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A089633 A362815 A358772 * A340856 A325100 A360007

Adjacent sequences: A003169 A003170 A003171 * A003173 A003174 A003175

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

The table in Borevich and Shafarevich extends to 497.

Status

approved