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.
Terms in A343001 that are not 9 or a prime.
Different from A343002 since a(31) = 819 = 7*9*13.
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
FORMULA
a(n) = sqrt(A343024(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).
819 is a term since 819^2 = 670761 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) isA343025(n) = my(L=factor(n), w=omega(n)); 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 2 associated cyclic cubic fields: A343024, this sequence.
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Apr 02 2021
STATUS
approved