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

%I A003172 M0618 #42 Jun 08 2017 16:38:31
%S A003172 2,3,5,6,7,11,13,14,17,19,21,22,23,29,31,33,37,38,41,43,46,47,53,57,
%T A003172 59,61,62,67,69,71,73,77,83,86,89,93,94,97,101,103,107,109,113,118,
%U A003172 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
%N A003172 Q(sqrt n) is a unique factorization domain (or simple quadratic field).
%C A003172 Squarefree numbers n such that A003649 is 1. - _T. D. Noe_, Apr 02 2008
%D A003172 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 422-423.
%D A003172 E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields. British Association Mathematical Tables, Vol. 4, London, 1934. (See p. 1.)
%D A003172 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A003172 H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 296.
%H A003172 Charles R Greathouse IV, <a href="/A003172/b003172.txt">Table of n, a(n) for n = 1..10000</a> [terms 1 through 1000 by T. D. Noe]
%H A003172 A. M. Odlyzko, <a href="/A003246/a003246.pdf">Letters to N. J. A. Sloane Feb 1974</a>
%H A003172 R. G. Underwood, <a href="http://dx.doi.org/10.3390/axioms2010001">On the content bound for real quadratic field extensions</a>, Axioms 2013, 2, 1-9; doi:10.3390/axioms2010001.
%H A003172 <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%t A003172 Select[Range[2, 199], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[#]] == 1 &] (* _Alonso del Arte_, Apr 17 2015 *)
%o A003172 (PARI)
%o A003172 A007947(n)={my(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]); }
%o A003172 { for (n=2, 10^3,
%o A003172     if ( n!=A007947(n), next() );
%o A003172     K = bnfinit(x^2 - n);
%o A003172     if ( K.cyc == [], print1( n, ", ") );
%o A003172 ); }
%o A003172 /* _Joerg Arndt_, Oct 18 2012 */
%o A003172 (PARI) is(n)=issquarefree(n) && qfbclassno(if(n%4>1, 4, 1)*n)==1 \\ _Charles R Greathouse IV_, Jan 19 2017
%Y A003172 Cf. A061574 (includes negative n), A029702-A029705, A218038-A218042.
%K A003172 nonn,nice
%O A003172 1,1
%A A003172 _N. J. A. Sloane_
%E A003172 The table in Borevich and Shafarevich extends to 497.