OFFSET

1,1

COMMENTS

Numbers k such that k^2 is in A006832.

Numbers k such that A160498(k) >= 2.

Each term k is associated with A343003(k) cyclic cubic fields.

Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be 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), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.

LINKS

Jianing Song, Table of n, a(n) for n = 1..3200

LMFDB, Cubic fields

Wikipedia, Cubic field

Ka Lun Wong, Maximal Unramified Extensions of Cyclic Cubic Fields, (2011), Theses and Dissertations, 2781.

FORMULA

a(n) = sqrt(A343001(n)).

EXAMPLE

7 is a term since 7^2 = 49 is the discriminant of the cyclic cubic field Q[x]/(x^3 - x^2 - 2*x + 1).

9 is a term since 9^2 = 81 is the discriminant of the cyclic cubic field Q[x]/(x^3 - 3*x - 1).

PROG

(PARI) isA343001(n) = my(L=factor(n), w=omega(n)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); (n>1)

CROSSREFS

Discriminants and their square roots of cyclic cubic fields:

At least 1 associated cyclic cubic field: A343000, this sequence.

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Apr 02 2021

STATUS

approved