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A343000 Discriminants of cyclic cubic fields. 8
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 18 09:26 EST 2022. Contains 350454 sequences. (Running on oeis4.)