

A343003


Numbers k such that there are exactly 2 cyclic cubic fields with discriminant k^2.


7



63, 91, 117, 133, 171, 217, 247, 259, 279, 301, 333, 387, 403, 427, 469, 481, 511, 549, 553, 559, 589, 603, 657, 679, 703, 711, 721, 763, 793, 817, 871, 873, 889, 927, 949, 973, 981, 1027, 1057, 1099, 1141, 1143, 1147, 1159, 1251, 1261, 1267, 1273, 1333
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

It makes no difference if the word "cyclic" is omitted from the title because a cubic field is cyclic if and only if its discriminant is a square.
Numbers k such that A160498(k) = 4.
Numbers 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.
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) = sqrt(A343002(n)).


EXAMPLE

63 is a term since 63^2 = 3969 is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3  21x  28) and Q[x]/(x^3  21x  35).
91 is a term since 91^2 = 8281 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) isA343003(n) = if(omega(n)==2, if(n==63, 1, my(L=factor(n)); L[2, 1]%3==1 && L[2, 2]==1 && ((L[1, 1]%3==1 && L[1, 2]==1)  L[1, 1]^L[1, 2] == 9)), 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: A343002, this sequence.
Cf. A006832, A160498, A343023.
Sequence in context: A193417 A062375 A343025 * A253021 A039480 A023688
Adjacent sequences: A343000 A343001 A343002 * A343004 A343005 A343006


KEYWORD

nonn,easy


AUTHOR

Jianing Song, Apr 02 2021


STATUS

approved



