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A343023 Number of cyclic cubic fields with discriminant n^2.

%I A343023 #18 Apr 04 2021 00:57:30
%S A343023 0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%T A343023 0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,1,0,
%U A343023 0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0
%N A343023 Number of cyclic cubic fields with discriminant n^2.
%C A343023 Equivalently, number of cubic fields with discriminant n^2. That is to say, it makes no difference if the word "cyclic" is omitted from the title.
%C A343023 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.
%C A343023 Each term is 0 or a power of 2.
%C A343023 The first occurrence of 2^t is 9*A121940(t) for t >= 1.
%H A343023 Jianing Song, <a href="/A343023/b343023.txt">Table of n, a(n) for n = 1..16000</a>
%H A343023 LMFDB, <a href="https://www.lmfdb.org/NumberField/?degree=3">Cubic fields</a>
%H A343023 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Cubic_field">Cubic field</a>
%H A343023 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 A343023 a(n) = A160498(n)/2 for n > 1.
%e A343023 a(7) = 1 since there is only 1 (cyclic) cubic field with discriminant 7^2 = 49 is Q[x]/(x^3 - x^2 + x + 1).
%e A343023 a(63) = 2 since there are 2 (cyclic) cubic fields with discriminant 63^2 = 3969: Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
%e A343023 a(819) = 4 since there are 4 (cyclic) cubic fields with discriminant 819^2 = 670761: 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).
%e A343023 a(35) = 0 since it is not of form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3). Indeed, there are no (cyclic) cubic fields with discriminant 35^2 = 1225.
%o A343023 (PARI) a(n) = if(n<=1, 0, 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))); 2^(w-1))
%Y A343023 Cf. A160498, A121940, A343000 (discriminants of cyclic cubic fields), A343001 (indices of positive terms).
%K A343023 nonn,easy
%O A343023 1,63
%A A343023 _Jianing Song_, Apr 02 2021