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A343024 Discriminants with at least 2 associated cyclic cubic fields. 7
3969, 8281, 13689, 17689, 29241, 47089, 61009, 67081, 77841, 90601, 110889, 149769, 162409, 182329, 219961, 231361, 261121, 301401, 305809, 312481, 346921, 363609, 431649, 461041, 494209, 505521, 519841, 582169, 628849, 667489, 670761, 758641, 762129, 790321, 859329, 900601, 946729, 962361 (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) >= 4.

Terms in A343000 that are not 81 or a square of a prime.

Different from A343002 since a(31) = 819^2 = (7*9*13)^2.

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..1600

LMFDB, Cubic fields

Wikipedia, Cubic field

FORMULA

a(n) = A343025(n)^2.

EXAMPLE

8281 = 91^2 is a term since it is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - x^2 - 30x + 64) and Q[x]/(x^3 - x^2 - 30x - 27).

670761 = 819^2 is a term since it is the discriminant of the 4 cyclic cubic fields Q[x]/(x^3 - 273x - 91), Q[x]/(x^3 - 273x - 728), Q[x]/(x^3 - 273x - 1547) and Q[x]/(x^3 - 273x - 1729).

PROG

(PARI) isA343024(n) = if(issquare(n), my(k=sqrtint(n), L=factor(k), w=omega(k)); if(w<2, return(0)); 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: A343000, A343001.

Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.

At least 2 associated cyclic cubic fields: this sequence, A343025.

Exactly 2 associated cyclic cubic fields: A343002, A343003.

Cf. A006832, A160498, A343023.

Sequence in context: A163111 A045247 A329786 * A343002 A230067 A031561

Adjacent sequences:  A343021 A343022 A343023 * A343025 A343026 A343027

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Apr 02 2021

STATUS

approved

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Last modified November 27 20:49 EST 2021. Contains 349395 sequences. (Running on oeis4.)