A002052 -id:A002052 - cp's OEIS frontend

A260335 Prime determinants of forms with class number > 2.

79, 223, 359, 439, 443, 499, 659, 727, 839, 1087, 1091, 1171, 1223, 1327, 1367, 1523, 1567, 1627, 1787, 1811, 1847, 1907, 1987, 2027, 2099, 2143, 2207, 2251, 2399, 2459, 2467, 2543, 2659, 2711, 2971, 3023, 3163, 3251, 3391, 3719, 3739, 3803, 3967, 4139, 4159, 4271

Offset: 1

N. J. A. Sloane, Jul 27 2015

Also primes p == 3 (mod 4) such that Z[sqrt(p)] = Z[x]/(x^2 - p) is not a unique factorization domain (or equivalently, not a principal ideal domain). - Jianing Song, Feb 17 2021

M. Suryanarayana, Positive determinants of binary quadratic forms whose class-number is 2, Proceedings of the Indian Academy of Sciences. Section A, 2 (1935), 178-179.

Cf. A002052.

isA260335(p) = isprime(p) && (p%4==3) && quadclassunit(4*p)[1] > 1 \\ Jianing Song, Feb 17 2021

More terms from Jianing Song, Feb 17 2021

A341649 Integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a unique factorization domain.

Original entry on oeis.org

-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

Offset: 1

Jianing Song, Feb 16 2021

sign

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.

			Z[sqrt(-1)] = Z[i] is the ring of Gaussian integers, which is a unique factorization domain.

Jianing Song, Table of n, a(n) for n = 1..10000
Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Mathematics Stack Exchange, Unique factorization domain that is not a Principal ideal domain
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A002052 (odd primes in this sequence).

isA341649(n) = my(D=4*n); isfundamental(D) && quadclassunit(D)[1] == 1

cp's OEIS Frontend

A260335 Prime determinants of forms with class number > 2.

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