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Number of conjugacy classes of primitive elements in GF(13^n) which have trace 0.
6

%I #15 Jun 02 2024 08:23:48

%S 0,0,18,112,1904,17184,229848,1686008,29713758

%N Number of conjugacy classes of primitive elements in GF(13^n) which have trace 0.

%C Also number of primitive polynomials of degree n over GF(13) whose second-highest coefficient is 0.

%F a(n) = A192216(n) / n.

%o (GAP)

%o p := 13;

%o a := function(n)

%o local q, k, cnt, x;

%o q:=p^n; k:=GF(p, n); cnt:=0;

%o for x in k do

%o if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then

%o cnt := cnt+1;

%o fi;

%o od;

%o return cnt/n;

%o end;

%o for n in [1..16] do Print (a(n), ", "); od;

%o (Sage) # See A192507 (change first line p=3 to p=13)

%Y Cf. A152049 (GF(2^n)), A192507 (GF(3^n)), A192508 (GF(5^n)), A192509 (GF(7^n)), A192510 (GF(11^n)).

%K nonn,hard,more

%O 1,3

%A _Joerg Arndt_, Jul 03 2011

%E a(7)-a(9) from _Robin Visser_, Jun 01 2024