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A343001 Square roots of discriminants of cyclic cubic fields. 8
7, 9, 13, 19, 31, 37, 43, 61, 63, 67, 73, 79, 91, 97, 103, 109, 117, 127, 133, 139, 151, 157, 163, 171, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 279, 283, 301, 307, 313, 331, 333, 337, 349, 367, 373, 379, 387, 397, 403, 409, 421, 427 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Numbers k such that k^2 is in A006832.

Numbers k such that A160498(k) >= 2.

Each term k 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) = sqrt(A343001(n)).

EXAMPLE

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

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

PROG

(PARI) isA343001(n) = my(L=factor(n), w=omega(n)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); (n>1)

CROSSREFS

Discriminants and their square roots of cyclic cubic fields:

At least 1 associated cyclic cubic field: A343000, this sequence.

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: A258616 A125866 A027692 * A297063 A185720 A032487

Adjacent sequences:  A342998 A342999 A343000 * A343002 A343003 A343004

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.)