|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
|
|
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
|
|
|
STATUS
|
approved
|
|
|
|