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A343003 Numbers k such that there are exactly 2 cyclic cubic fields with discriminant k^2.

%I A343003 #14 Apr 03 2021 12:09:43
%S A343003 63,91,117,133,171,217,247,259,279,301,333,387,403,427,469,481,511,
%T A343003 549,553,559,589,603,657,679,703,711,721,763,793,817,871,873,889,927,
%U A343003 949,973,981,1027,1057,1099,1141,1143,1147,1159,1251,1261,1267,1273,1333
%N A343003 Numbers k such that there are exactly 2 cyclic cubic fields with discriminant k^2.
%C A343003 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 A343003 Numbers k such that A160498(k) = 4.
%C A343003 Numbers of the form (i) 9p, where p is a prime congruent to 1 modulo 3; (ii) pq, where p, q are distinct primes congruent to 1 modulo 3.
%C A343003 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 A343003 Jianing Song, <a href="/A343003/b343003.txt">Table of n, a(n) for n = 1..10000</a>
%H A343003 LMFDB, <a href="https://www.lmfdb.org/NumberField/?degree=3">Cubic fields</a>
%H A343003 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Cubic_field">Cubic field</a>
%F A343003 a(n) = sqrt(A343002(n)).
%e A343003 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 A343003 91 is a term since 91^2 = 8281 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).
%o A343003 (PARI) isA343003(n) = if(omega(n)==2, if(n==63, 1, my(L=factor(n)); L[2,1]%3==1 && L[2,2]==1 && ((L[1,1]%3==1 && L[1,2]==1) || L[1,1]^L[1,2] == 9)), 0)
%Y A343003 Discriminants and their square roots of cyclic cubic fields:
%Y A343003 At least 1 associated cyclic cubic field: A343000, A343001.
%Y A343003 Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
%Y A343003 At least 2 associated cyclic cubic fields: A343024, A343025.
%Y A343003 Exactly 2 associated cyclic cubic fields: A343002, this sequence.
%Y A343003 Cf. A006832, A160498, A343023.
%K A343003 nonn,easy
%O A343003 1,1
%A A343003 _Jianing Song_, Apr 02 2021