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%I #20 Nov 20 2021 20:47:14
%S 7,9,13,19,31,37,43,61,63,67,73,79,91,97,103,109,117,127,133,139,151,
%T 157,163,171,181,193,199,211,217,223,229,241,247,259,271,277,279,283,
%U 301,307,313,331,333,337,349,367,373,379,387,397,403,409,421,427
%N Square roots of discriminants of cyclic cubic fields.
%C Numbers k such that k^2 is in A006832.
%C Numbers k such that A160498(k) >= 2.
%C Each term k is associated with A343003(k) cyclic cubic fields.
%C Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be 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), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.
%H Jianing Song, <a href="/A343001/b343001.txt">Table of n, a(n) for n = 1..3200</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>
%H Ka Lun Wong, <a href="https://scholarsarchive.byu.edu/etd/2781">Maximal Unramified Extensions of Cyclic Cubic Fields</a>, (2011), Theses and Dissertations, 2781.
%F a(n) = sqrt(A343001(n)).
%e 7 is a term since 7^2 = 49 is the discriminant of the cyclic cubic field Q[x]/(x^3 - x^2 - 2*x + 1).
%e 9 is a term since 9^2 = 81 is the discriminant of the cyclic cubic field Q[x]/(x^3 - 3*x - 1).
%o (PARI) isA343001(n) = my(L=factor(n), w=omega(n)); for(i=1, w, if(!((L[i,1]%3==1 && L[i,2]==1) || L[i,1]^L[i,2] == 9), return(0))); (n>1)
%Y Discriminants and their square roots of cyclic cubic fields:
%Y At least 1 associated cyclic cubic field: A343000, this sequence.
%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: A343002, A343003.
%Y Cf. A006832, A160498, A343023.
%K nonn,easy
%O 1,1
%A _Jianing Song_, Apr 02 2021
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