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A219361 Positive integers n such that the ring of integers of Q(sqrt n) is a UFD.

%I A219361 #40 Nov 01 2014 13:01:15
%S A219361 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,
%T A219361 31,32,33,36,37,38,41,43,44,45,46,47,48,49,50,52,53,54,56,57,59,61,62,
%U A219361 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
%N A219361 Positive integers n such that the ring of integers of Q(sqrt n) is a UFD.
%C A219361 A003172 is the main entry for this sequence, which removes duplicates (i.e., for nonsquarefree n) like Q(sqrt(8)) = Q(sqrt(2)).
%C A219361 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
%H A219361 Charles R Greathouse IV, <a href="/A219361/b219361.txt">Table of n, a(n) for n = 1..10000</a>
%e A219361 The following are in this sequence:
%e A219361   1, 4, 9, 16, ... because Z is a UFD (by the Fundamental Theorem of Arithmetic);
%e A219361   2, 8, 18, 32, ... because Z[sqrt(2)] has unique factorization;
%e A219361   3, 12, 27, 48, ... because Z[(1+sqrt(3))/2] has unique factorization;
%e A219361   5, 20, 45, 80, ... because Z[(1+sqrt(5))/2] has unique factorization.
%t A219361 Select[Range[100], NumberFieldClassNumber[Sqrt[#]] == 1 &] (* _Alonso del Arte_, Feb 19 2013 *)
%o A219361 (PARI) is(n)=n=core(n); n==1 || !#bnfinit('x^2-n).cyc
%Y A219361 Cf. A029702, A061574.
%K A219361 nonn
%O A219361 1,2
%A A219361 _Charles R Greathouse IV_, Feb 19 2013