A343000 Discriminants of cyclic cubic fields.

49, 81, 169, 361, 961, 1369, 1849, 3721, 3969, 4489, 5329, 6241, 8281, 9409, 10609, 11881, 13689, 16129, 17689, 19321, 22801, 24649, 26569, 29241, 32761, 37249, 39601, 44521, 47089, 49729, 52441, 58081, 61009, 67081, 73441, 76729, 77841, 80089, 90601, 94249, 97969

Offset

1,1

Comments

Square terms in A006832.

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

Each term k^2 is associated with A343003(k) cyclic cubic fields.

Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be 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), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.

Links

Jianing Song, Table of n, a(n) for n = 1..3200

LMFDB, Cubic fields

Wikipedia, Cubic field

Ka Lun Wong, Maximal Unramified Extensions of Cyclic Cubic Fields, (2011), Theses and Dissertations, 2781.

Formula

a(n) = A343001(n)^2.

Example

49 = 7^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - x^2 + x + 1).
81 = 9^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - 3x - 1).

Prog

(PARI) isA343000(n) = if(issquare(n)&&n>1, my(k=sqrtint(n), L=factor(k), w=omega(k)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 1)

Crossrefs

Discriminants and their square roots of cyclic cubic fields:

At least 1 associated cyclic cubic field: this sequence, A343001.

Exactly 1 associated cyclic cubic field: A343022, 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: A286095 A106311 A006832 * A343022 A250074 A247678

Adjacent sequences: A342997 A342998 A342999 * A343001 A343002 A343003

Keyword

nonn,easy

Author

Jianing Song, Apr 02 2021

Status

approved