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A192508 Number of conjugacy classes of primitive elements in GF(5^n) which have trace 0.

%I A192508 #14 May 10 2024 08:52:20
%S A192508 0,0,4,8,54,140,1116,2976,19828,58388,443892,1036180,9390024,27996724,
%T A192508 175396812
%N A192508 Number of conjugacy classes of primitive elements in GF(5^n) which have trace 0.
%C A192508 Also number of primitive polynomials of degree n over GF(5) whose second-highest coefficient is 0.
%F A192508 a(n) = A192213(n) / n
%o A192508 (GAP)
%o A192508 p := 5;
%o A192508 a := function(n)
%o A192508     local q, k, cnt, x;
%o A192508     q:=p^n;  k:=GF(p, n);  cnt:=0;
%o A192508     for x in k do
%o A192508         if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
%o A192508             cnt := cnt+1;
%o A192508         fi;
%o A192508     od;
%o A192508     return cnt/n;
%o A192508 end;
%o A192508 for n in [1..16] do  Print (a(n), ", ");  od;
%o A192508 (Sage) # See A192507 (change first line p=3 to p=5)
%Y A192508 Cf. A152049 (GF(2^n)), A192507 (GF(3^n)), A192509 (GF(7^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).
%K A192508 nonn,hard,more
%O A192508 1,3
%A A192508 _Joerg Arndt_, Jul 03 2011
%E A192508 Added terms 19828..443892, _Joerg Arndt_, Oct 03 2012
%E A192508 a(12)-a(15) from _Robin Visser_, May 10 2024