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A343025 Numbers k such that there are at least 2 cyclic cubic fields with discriminant k^2.

%I A343025 #15 Apr 03 2021 16:18:44
%S A343025 63,91,117,133,171,217,247,259,279,301,333,387,403,427,469,481,511,
%T A343025 549,553,559,589,603,657,679,703,711,721,763,793,817,819,871,873,889,
%U A343025 927,949,973,981,1027,1057,1099,1141,1143,1147,1159,1197,1251,1261,1267
%N A343025 Numbers k such that there are at least 2 cyclic cubic fields with discriminant k^2.
%C A343025 It makes no difference if the word "cyclic" is omitted from the title because a cubic field is cyclic if and only if its discriminant is a square.
%C A343025 Numbers k such that A160498(k) >= 4.
%C A343025 Terms in A343001 that are not 9 or a prime.
%C A343025 Different from A343002 since a(31) = 819 = 7*9*13.
%C A343025 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 A343025 Jianing Song, <a href="/A343025/b343025.txt">Table of n, a(n) for n = 1..1600</a>
%H A343025 LMFDB, <a href="https://www.lmfdb.org/NumberField/?degree=3">Cubic fields</a>
%H A343025 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Cubic_field">Cubic field</a>
%F A343025 a(n) = sqrt(A343024(n)).
%e A343025 63 is a term since 63^2 = 3969 is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
%e A343025 819 is a term since 819^2 = 670761 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 A343025 (PARI) isA343025(n) = my(L=factor(n), w=omega(n)); 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 A343025 Discriminants and their square roots of cyclic cubic fields:
%Y A343025 At least 1 associated cyclic cubic field: A343000, A343001.
%Y A343025 Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
%Y A343025 At least 2 associated cyclic cubic fields: A343024, this sequence.
%Y A343025 Exactly 2 associated cyclic cubic fields: A343002, A343003.
%Y A343025 Cf. A006832, A160498, A343023.
%K A343025 nonn,easy
%O A343025 1,1
%A A343025 _Jianing Song_, Apr 02 2021