A343022 Discriminants with exactly 1 associated cyclic cubic field.

49, 81, 169, 361, 961, 1369, 1849, 3721, 4489, 5329, 6241, 9409, 10609, 11881, 16129, 19321, 22801, 24649, 26569, 32761, 37249, 39601, 44521, 49729, 52441, 58081, 73441, 76729, 80089, 94249, 97969, 109561, 113569, 121801, 134689, 139129, 143641, 157609, 167281, 177241

Offset

1,1

Comments

A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.

Numbers of the form k^2 where A160498(k) = 2.

Numbers of the form k^2 where k is in A002476 U {9}. That is to say, numbers of the form k^2 where k = 9 or is a prime congruent to 1 modulo 3.

In general, there are exactly 2^(t-1) (cyclic) cubic fields with discriminant k^2 if and only if k is of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), See A343000 for more detailed information.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

LMFDB, Cubic fields

Wikipedia, Cubic field

Formula

a(n) = A002476(n-1)^2 for n >= 3.

Example

169 is a term since the one (and only one) cyclic cubic field with that discriminant is Q[x]/(x^3 - x^2 - 4x - 1).

Prog

(PARI) isA343022(n) = if(issquare(n), my(k=sqrtint(n)); k==9 || (isprime(k) && k%3==1), 0)

Crossrefs

Discriminants and their square roots of cyclic cubic fields:

At least 1 associated cyclic cubic field: A343000, A343001.

Exactly 1 associated cyclic cubic field: this sequence, A002476 U {9}.

At least 2 associated cyclic cubic fields: A343024, A343025.

Exactly 2 associated cyclic cubic fields: A343002, A343003.

Cf. A006832, A160498, A343023.

Sequence in context: A106311 A006832 A343000 * A250074 A247678 A093894

Adjacent sequences: A343019 A343020 A343021 * A343023 A343024 A343025

Keyword

nonn,easy

Author

Jianing Song, Apr 02 2021

Status

approved