

A343001


Square roots of discriminants of cyclic cubic fields.


8



7, 9, 13, 19, 31, 37, 43, 61, 63, 67, 73, 79, 91, 97, 103, 109, 117, 127, 133, 139, 151, 157, 163, 171, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 279, 283, 301, 307, 313, 331, 333, 337, 349, 367, 373, 379, 387, 397, 403, 409, 421, 427
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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_{t1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t1) = 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.
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, A343003.
Cf. A006832, A160498, A343023.
Sequence in context: A258616 A125866 A027692 * A297063 A185720 A032487
Adjacent sequences: A342998 A342999 A343000 * A343002 A343003 A343004


KEYWORD

nonn,easy


AUTHOR

Jianing Song, Apr 02 2021


STATUS

approved



