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%I #15 Apr 04 2021 00:57:38
%S 3969,8281,13689,17689,29241,47089,61009,67081,77841,90601,110889,
%T 149769,162409,182329,219961,231361,261121,301401,305809,312481,
%U 346921,363609,431649,461041,494209,505521,519841,582169,628849,667489,670761,758641,762129,790321,859329,900601,946729,962361
%N Discriminants with at least 2 associated cyclic cubic fields.
%C A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.
%C Numbers of the form k^2 where A160498(k) >= 4.
%C Terms in A343000 that are not 81 or a square of a prime.
%C Different from A343002 since a(31) = 819^2 = (7*9*13)^2.
%C In general, there are exactly 2^(t-1) (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_{t-1}) with distinct primes p_i == 1 (mod 3), See A343000 for more detailed information.
%H Jianing Song, <a href="/A343024/b343024.txt">Table of n, a(n) for n = 1..1600</a>
%H LMFDB, <a href="https://www.lmfdb.org/NumberField/?degree=3">Cubic fields</a>
%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Cubic_field">Cubic field</a>
%F a(n) = A343025(n)^2.
%e 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).
%e 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).
%o (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)
%Y Discriminants and their square roots of cyclic cubic fields:
%Y At least 1 associated cyclic cubic field: A343000, A343001.
%Y Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
%Y At least 2 associated cyclic cubic fields: this sequence, A343025.
%Y Exactly 2 associated cyclic cubic fields: A343002, A343003.
%Y Cf. A006832, A160498, A343023.
%K nonn,easy
%O 1,1
%A _Jianing Song_, Apr 02 2021
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