A029702 -id:A029702 - cp's OEIS frontend

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

Offset: 1

Charles R Greathouse IV, Feb 19 2013

nonn

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

			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.

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

Cf. A029702, A061574.

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

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

A081363 Smallest squarefree integer k such that Q(sqrt(k)) has class number n.

Original entry on oeis.org

2, 10, 79, 82, 401, 235, 577, 226, 1129, 1111, 1297, 730, 4759, 1534, 9871, 2305, 7054, 4954, 15409, 3601, 7057, 4762, 23593, 9634, 24859, 13321, 8761, 5626, 49281, 11665, 97753, 15130, 55339, 19882, 25601, 18226, 24337, 19834, 41614, 16899, 55966, 47959

Offset: 1

Dean Hickerson, Mar 19 2003

nonn

What is known about the asymptotics of this sequence? - Charles R Greathouse IV, Jan 26 2017

Records: 2, 10, 79, 82, 401, 577, 1129, 1297, 4759, 9871, 15409, 23593, 24859, 49281, 97753, 106537, 159199, 197137, 212137, 239119, 245023, 444089, 589822, 614849, 815413, 837929, 943951, 1025494, 1224121, 1240369, 1333255, 1334026, ..., . - Robert G. Wilson v, Apr 12 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..214

Cf. A081319, A081364, A094619, A094612, A094613, A094614.

Cf. A219361, A029702, A029703, A029704, A029705, A218038, A218039, A218040, A218041, A218042, A276541. - Robert G. Wilson v, Apr 12 2017

t[] := 0; k = 2; While[k < 56000, cn = NumberFieldClassNumber@Sqrt@k; If[ t[cn] == 0, t[cn] = k; Print[{cn, k}]]; k++]; t@# & /@ Range@ 42 (* _Robert G. Wilson v, Apr 12 2017 *)

v=vector(100); for(n=2,1e6, if(!issquarefree(n), next); t=qfbclassno(if(n%4>1,4*n,n)); if(t<=#v && v[t]==0, v[t]=n; print("a("t") = "n))) \\ Charles R Greathouse IV, Jan 26 2017

More terms from Max Alekseyev, Apr 28 2010

cp's OEIS Frontend

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

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