login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A341649 Integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a unique factorization domain. 1
-2, -1, 2, 3, 6, 7, 11, 14, 19, 22, 23, 31, 38, 43, 46, 47, 59, 62, 67, 71, 83, 86, 94, 103, 107, 118, 127, 131, 134, 139, 151, 158, 163, 166, 167, 179, 191, 199, 206, 211, 214, 227, 239, 251, 262, 263, 271, 278, 283, 302, 307, 311, 331, 334, 347, 358, 367, 379 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Equivalently, integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a principal ideal domain.
-2, -1, together with k such that 4*k is in A003656.
All terms are squarefree and congruent to 2 or 3 modulo 4. It appears that the terms > 2 are of the form p or 2*p, where p is a prime congruent to 3 modulo 4. [This is correct; see Theorem 1 and Theorem 2 of Ezra Brown's link. - Jianing Song, Feb 24 2021]
The smallest prime p == 3 (mod 4) that is not a term is p = 79. The smallest prime p == 3 (mod 4) such that 2*p is not a term is p = 71.
LINKS
Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Eric Weisstein's World of Mathematics, Class Number
EXAMPLE
Z[sqrt(-1)] = Z[i] is the ring of Gaussian integers, which is a unique factorization domain.
PROG
(PARI) isA341649(n) = my(D=4*n); isfundamental(D) && quadclassunit(D)[1] == 1
CROSSREFS
Cf. A003656, A002052 (odd primes in this sequence).
Sequence in context: A108618 A097719 A249050 * A056493 A289352 A277619
KEYWORD
sign
AUTHOR
Jianing Song, Feb 16 2021
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 14 19:09 EDT 2024. Contains 374323 sequences. (Running on oeis4.)