A146209 -id:A146209 - cp's OEIS frontend

A029702 Q(sqrt(n)) has class number 2.

10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 65, 66, 70, 74, 78, 85, 87, 91, 95, 102, 105, 106, 110, 111, 114, 115, 119, 122, 123, 138, 143, 146, 154, 155, 159, 165, 174, 178, 182, 183, 185, 186, 187, 190, 194, 202, 203, 205, 215, 218, 221, 222, 230, 238, 246

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

nonn

Smallest term that is in A146209 but not this sequence is 79, since Q(sqrt(79)) has class number 3. - Alonso del Arte, Aug 25 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to quadratic fields

Cf. A003172, A029703-A029705, A218038-A218042.

Select[Range[246], SquareFreeQ[#] && NumberFieldClassNumber@Sqrt[#] == 2 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

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 == [2], print1( n, ", ") );
); }
/* Joerg Arndt, Oct 18 2012 */

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

Original entry on oeis.org

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

A278837 Primes p such that the ring of algebraic integers of Q(sqrt(p)) does not have unique factorization.

Original entry on oeis.org

79, 223, 229, 257, 359, 401, 439, 443, 499, 577, 659, 727, 733, 761, 839, 1009, 1087, 1091, 1093, 1129, 1171, 1223, 1229, 1297, 1327, 1367, 1373, 1429, 1489, 1523, 1567, 1601, 1627, 1787, 1811, 1847, 1901, 1907, 1987, 2027, 2029, 2081, 2089, 2099, 2143, 2153, 2207, 2213, 2251, 2399, 2459, 2467

Offset: 1

Alonso del Arte, Nov 28 2016

nonn

It is still unknown whether there are infinitely many real, positive, squarefree d such that O_(Q(sqrt(d))) has unique factorization (or, to put it another way, the class number is 1).
If one only looks at small prime numbers, one could easily be tempted to think that if p is prime then O_(Q(sqrt(p))) has unique factorization.
By contrast, given distinct primes p and q, one could think that O_(Q(sqrt(p*q))) generally does not have unique factorization, especially if p = 5.
It then often happens that both p and q are irreducible, and therefore p*q = (sqrt(p*q))^2 represents two distinct factorizations of the same number.
Such an obvious example of multiple distinct factorizations is obviously not available in O_(Q(sqrt(p))).

			In Z[sqrt(79)], to pick just one example of a number having more than one distinct factorization, we verify that 3 and 5 are both irreducible, yet 15 = 3 * 5 = (-1)*(8 - sqrt(79))*(8 + sqrt(79)). Thus 79 is in the sequence.
Z[sqrt(83)] is a unique factorization domain, hence 83 is not in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A146209.

Select[Prime[Range[100]], NumberFieldClassNumber[Sqrt[#]] > 1 &]

Missing term 2089 added by Emmanuel Vantieghem, Mar 08 2019

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A029702 Q(sqrt(n)) has class number 2.

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

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PARI

A278837 Primes p such that the ring of algebraic integers of Q(sqrt(p)) does not have unique factorization.

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