A219361 - OEIS

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A219361

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

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, 31, 32, 33, 36, 37, 38, 41, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 59, 61, 62, 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

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OFFSET

1,2

COMMENTS

A003172 is the main entry for this sequence, which removes duplicates (i.e., for nonsquarefree n) like Q(sqrt(8)) = Q(sqrt(2)).

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

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

EXAMPLE

The following are in this sequence:

1, 4, 9, 16, ... because Z is a UFD (by the Fundamental Theorem of Arithmetic);

2, 8, 18, 32, ... because Z[sqrt(2)] has unique factorization;

3, 12, 27, 48, ... because Z[(1+sqrt(3))/2] has unique factorization;

5, 20, 45, 80, ... because Z[(1+sqrt(5))/2] has unique factorization.

MATHEMATICA

Select[Range[100], NumberFieldClassNumber[Sqrt[#]] == 1 &] (* Alonso del Arte, Feb 19 2013 *)

PROG

(PARI) is(n)=n=core(n); n==1 || !#bnfinit('x^2-n).cyc