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

Author

Jianing Song, Apr 02 2021

Keywords

• nonn
• easy

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.

Examples

			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).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• LMFDB, Cubic fields
• Wikipedia, Cubic field

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.

Programs

PARI

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

Formula

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