A192509 - OEIS

%I #22 Jun 02 2024 11:04:54

%S 0,0,3,20,160,846,5426,27360,196740,1215548,8552408,37330020

%N Number of conjugacy classes of primitive elements in GF(7^n) which have trace 0.

%C Also number of primitive polynomials of degree n over GF(7) whose second-highest coefficient is 0.

%F a(n) = A192214(n) / n.

%o (GAP)

%o p := 7;

%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=7)

%Y Cf. A152049 (GF(2^n)), A192507 (GF(3^n)), A192508 (GF(5^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).

%K nonn,hard,more

%O 1,3

%A _Joerg Arndt_, Jul 03 2011

%E a(7)-a(9) from _Joerg Arndt_, Oct 14 2012

%E a(10)-a(12) from _Robin Visser_, Jun 01 2024