

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^(t1) (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_{t1}) 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



