A219361 -id:A219361 - cp's OEIS frontend

A146209 Integers a(n) for which the factorization in the real quadratic field Q(sqrt(a(n))) is not unique.

10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 65, 66, 70, 74, 78, 79, 82, 85, 87, 91, 95, 102, 105, 106, 110, 111, 114, 115, 119, 122, 123, 130, 138, 142, 143, 145, 146, 154, 155, 159, 165, 170, 174, 178, 182, 183, 185, 186, 187, 190, 194, 195

Offset: 1

Pahikkala Jussi, Oct 28 2008

nonn

The class number of Q(sqrt(a(n))) is greater than 1.

Contains A029702, A053330 and A051990 as subsequences. See A219361 for positive integers D for which Q(sqrt D) is a UFD. - M. F. Hasler, Oct 30 2014

			For n = 6, a(6) = 35 since 35 is the sixth positive squarefree integer u for which the factorization in Q(sqrt(u)) is not unique.

Z. I. Borevich and I. R. Shafarevich, Zahlentheorie. Birkhäuser Verlag, Basel und Stuttgart (1966).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003172.

Select[Range[200], SquareFreeQ[#] && NumberFieldClassNumber[Sqrt[#]] > 1 &] (* Alonso del Arte, Sep 05 2012 *)

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

A276541 Positive integers n such that the ring of integers of Q(sqrt(n)) is not a UFD.

Original entry on oeis.org

10, 15, 26, 30, 34, 35, 39, 40, 42, 51, 55, 58, 60, 65, 66, 70, 74, 78, 79, 82, 85, 87, 90, 91, 95, 102, 104, 105, 106, 110, 111, 114, 115, 119, 120, 122, 123, 130, 135, 136, 138, 140, 142, 143, 145, 146, 154, 155, 156, 159, 160, 165, 168, 170, 174, 178, 182

Offset: 1

Robert G. Wilson v, Apr 09 2017

nonn

Complement of A219361.

Cf. A219361.

Select[Range[210], NumberFieldClassNumber@Sqrt@# > 1 &]

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A146209 Integers a(n) for which the factorization in the real quadratic field Q(sqrt(a(n))) is not unique.

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

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A276541 Positive integers n such that the ring of integers of Q(sqrt(n)) is not a UFD.

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