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A343022 Discriminants with exactly 1 associated cyclic cubic field.

%I A343022 #14 Apr 03 2021 12:10:30
%S A343022 49,81,169,361,961,1369,1849,3721,4489,5329,6241,9409,10609,11881,
%T A343022 16129,19321,22801,24649,26569,32761,37249,39601,44521,49729,52441,
%U A343022 58081,73441,76729,80089,94249,97969,109561,113569,121801,134689,139129,143641,157609,167281,177241
%N A343022 Discriminants with exactly 1 associated cyclic cubic field.
%C A343022 A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.
%C A343022 Numbers of the form k^2 where A160498(k) = 2.
%C A343022 Numbers of the form k^2 where k is in A002476 U {9}. That is to say, numbers of the form k^2 where k = 9 or is a prime congruent to 1 modulo 3.
%C A343022 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 A343022 Jianing Song, <a href="/A343022/b343022.txt">Table of n, a(n) for n = 1..10000</a>
%H A343022 LMFDB, <a href="https://www.lmfdb.org/NumberField/?degree=3">Cubic fields</a>
%H A343022 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Cubic_field">Cubic field</a>
%F A343022 a(n) = A002476(n-1)^2 for n >= 3.
%e A343022 169 is a term since the one (and only one) cyclic cubic field with that discriminant is Q[x]/(x^3 - x^2 - 4x - 1).
%o A343022 (PARI) isA343022(n) = if(issquare(n), my(k=sqrtint(n)); k==9 || (isprime(k) && k%3==1), 0)
%Y A343022 Discriminants and their square roots of cyclic cubic fields:
%Y A343022 At least 1 associated cyclic cubic field: A343000, A343001.
%Y A343022 Exactly 1 associated cyclic cubic field: this sequence, A002476 U {9}.
%Y A343022 At least 2 associated cyclic cubic fields: A343024, A343025.
%Y A343022 Exactly 2 associated cyclic cubic fields: A343002, A343003.
%Y A343022 Cf. A006832, A160498, A343023.
%K A343022 nonn,easy
%O A343022 1,1
%A A343022 _Jianing Song_, Apr 02 2021