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A343002 Discriminants with exactly 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, 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.

Numbers of the form k^2 where k is of the form (i) 9p, where p is a prime congruent to 1 modulo 3; (ii) pq, where p, q are distinct primes congruent to 1 modulo 3.

Products of two nonequal terms in A343022.

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..10000

LMFDB, Cubic fields

Wikipedia, Cubic field

FORMULA

a(n) = A343003(n)^2.

EXAMPLE

3969 = 63^2 is a term since it is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).

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).

PROG

(PARI) isA343002(n) = if(omega(n)==2, if(n==3969, 1, my(L=factor(n)); L[2, 1]%3==1 && L[2, 2]==2 && ((L[1, 1]%3==1 && L[1, 2]==2) || L[1, 1]^L[1, 2] == 81)), 0)

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: A343024, A343025.

Exactly 2 associated cyclic cubic fields: this sequence, A343003.

Cf. A006832, A160498, A343023.

Sequence in context: A045247 A329786 A343024 * A230067 A031561 A031741

Adjacent sequences:  A342999 A343000 A343001 * A343003 A343004 A343005

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Apr 02 2021

STATUS

approved

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Last modified December 2 17:56 EST 2021. Contains 349445 sequences. (Running on oeis4.)