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A343022 Discriminants with exactly 1 associated cyclic cubic field. 7
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 18 07:55 EST 2022. Contains 350454 sequences. (Running on oeis4.)