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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

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.