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A192507 Number of conjugacy classes of primitive elements in GF(3^n) which have trace 0.

%I A192507 #24 Apr 26 2024 12:25:50
%S A192507 0,0,1,2,7,14,52,104,333,870,2571,4590,20440,56736,133782,327558,
%T A192507 1265391,2612694,10188836,20769420,76562106
%N A192507 Number of conjugacy classes of primitive elements in GF(3^n) which have trace 0.
%C A192507 Also number of primitive polynomials of degree n over GF(3) whose second-highest coefficient is 0.
%F A192507 a(n) = A192212(n) / n.
%o A192507 (GAP)
%o A192507 p := 3;
%o A192507 a := function(n)
%o A192507     local q, k, cnt, x;
%o A192507     q:=p^n;  k:=GF(p, n);  cnt:=0;
%o A192507     for x in k do
%o A192507         if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
%o A192507             cnt := cnt+1;
%o A192507         fi;
%o A192507     od;
%o A192507     return cnt/n;
%o A192507 end;
%o A192507 for n in [1..16] do  Print (a(n), ", ");  od;
%o A192507 (Sage) # much more efficient
%o A192507 p=3; # choose characteristic
%o A192507 for n in range(1,66):
%o A192507     F = GF(p^n, 'x')
%o A192507     g = F.multiplicative_generator() # generator
%o A192507     vt = vector(ZZ,p) # stats: trace
%o A192507     m = p^n - 1 # size of multiplicative group
%o A192507     # Compute all irreducible polynomials via Lyndon words:
%o A192507     for w in LyndonWords(p,n): # digits of Lyndon words range form 1,..,p
%o A192507         e = sum( (w[j]-1) * p^j for j in range(0,n) )
%o A192507         if gcd(m, e) == 1: # primitive elements only
%o A192507             f = g^e
%o A192507             t = f.trace().lift(); # trace (over ZZ)
%o A192507             vt[t] += 1
%o A192507     print(vt[0]) # choose index 0,1,..,p-1 for different traces
%o A192507 # _Joerg Arndt_, Oct 03 2012
%Y A192507 Cf. A152049 (GF(2^n)), A192507 (GF(5^n)), A192509 (GF(7^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).
%Y A192507 Cf. A027385 (number of primitive polynomials of degree n over GF(3)).
%K A192507 nonn,hard,more
%O A192507 1,4
%A A192507 _Joerg Arndt_, Jul 03 2011
%E A192507 Added terms >=2571, _Joerg Arndt_, Oct 03 2012
%E A192507 a(18)-a(21) from _Robin Visser_, Apr 26 2024