

A343024


Discriminants with at least 2 associated cyclic cubic fields.


7



3969, 8281, 13689, 17689, 29241, 47089, 61009, 67081, 77841, 90601, 110889, 149769, 162409, 182329, 219961, 231361, 261121, 301401, 305809, 312481, 346921, 363609, 431649, 461041, 494209, 505521, 519841, 582169, 628849, 667489, 670761, 758641, 762129, 790321, 859329, 900601, 946729, 962361
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OFFSET

1,1


COMMENTS

A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.
Numbers of the form k^2 where A160498(k) >= 4.
Terms in A343000 that are not 81 or a square of a prime.
Different from A343002 since a(31) = 819^2 = (7*9*13)^2.
In general, there are exactly 2^(t1) (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_{t1}) with distinct primes p_i == 1 (mod 3), See A343000 for more detailed information.


LINKS

Jianing Song, Table of n, a(n) for n = 1..1600
LMFDB, Cubic fields
Wikipedia, Cubic field


FORMULA

a(n) = A343025(n)^2.


EXAMPLE

8281 = 91^2 is a term since it 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).
670761 = 819^2 is a term since it is the discriminant of the 4 cyclic cubic fields Q[x]/(x^3  273x  91), Q[x]/(x^3  273x  728), Q[x]/(x^3  273x  1547) and Q[x]/(x^3  273x  1729).


PROG

(PARI) isA343024(n) = if(issquare(n), my(k=sqrtint(n), L=factor(k), w=omega(k)); if(w<2, return(0)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1)  L[i, 1]^L[i, 2] == 9), return(0))); 1)


CROSSREFS

Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: A343000, A343001.
Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
At least 2 associated cyclic cubic fields: this sequence, A343025.
Exactly 2 associated cyclic cubic fields: A343002, A343003.
Cf. A006832, A160498, A343023.
Sequence in context: A163111 A045247 A329786 * A343002 A230067 A031561
Adjacent sequences: A343021 A343022 A343023 * A343025 A343026 A343027


KEYWORD

nonn,easy


AUTHOR

Jianing Song, Apr 02 2021


STATUS

approved



