A343002 Discriminants with exactly 2 associated cyclic cubic fields.

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

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