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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
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
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.
Sequence in context: A258616 A125866 A027692 * A297063 A185720 A032487
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Apr 02 2021
STATUS
approved