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

%I #14 May 10 2024 08:52:20

%S 0,0,4,8,54,140,1116,2976,19828,58388,443892,1036180,9390024,27996724,

%T 175396812

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

%C Also number of primitive polynomials of degree n over GF(5) whose second-highest coefficient is 0.

%F a(n) = A192213(n) / n

%o (GAP)

%o p := 5;

%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=5)

%Y Cf. A152049 (GF(2^n)), A192507 (GF(3^n)), A192509 (GF(7^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).

%K nonn,hard,more

%O 1,3

%A _Joerg Arndt_, Jul 03 2011

%E Added terms 19828..443892, _Joerg Arndt_, Oct 03 2012

%E a(12)-a(15) from _Robin Visser_, May 10 2024