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^(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(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:
Exactly 2 associated cyclic cubic fields: A343002, this sequence.
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Apr 02 2021
STATUS
approved