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%I #17 Apr 03 2021 12:13:42
%S 3969,8281,13689,17689,29241,47089,61009,67081,77841,90601,110889,
%T 149769,162409,182329,219961,231361,261121,301401,305809,312481,
%U 346921,363609,431649,461041,494209,505521,519841,582169,628849,667489,758641,762129,790321,859329,900601,946729,962361
%N Discriminants with exactly 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 Numbers of the form k^2 where k is 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 Products of two nonequal terms in A343022.
%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="/A343002/b343002.txt">Table of n, a(n) for n = 1..10000</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) = A343003(n)^2.
%e 3969 = 63^2 is a term since it is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
%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).
%o (PARI) isA343002(n) = if(omega(n)==2, if(n==3969, 1, my(L=factor(n)); L[2,1]%3==1 && L[2,2]==2 && ((L[1,1]%3==1 && L[1,2]==2) || L[1,1]^L[1,2] == 81)), 0)
%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: A343024, A343025.
%Y Exactly 2 associated cyclic cubic fields: this sequence, A343003.
%Y Cf. A006832, A160498, A343023.
%K nonn,easy
%O 1,1
%A _Jianing Song_, Apr 02 2021
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