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

0, 0, 4, 8, 54, 140, 1116, 2976, 19828, 58388, 443892, 1036180, 9390024, 27996724, 175396812

Offset: 1

Author

Joerg Arndt, Jul 03 2011

Keywords

• nonn
• hard
• more

Comments

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

Crossrefs

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

Programs

GAP

p := 5;
a := function(n)
    local q, k, cnt, x;
    q:=p^n;  k:=GF(p, n);  cnt:=0;
    for x in k do
        if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
            cnt := cnt+1;
        fi;
    od;
    return cnt/n;
end;
for n in [1..16] do  Print (a(n), ", ");  od;

Sage

# See A192507 (change first line p=3 to p=5)

Formula

a(n) = A192213(n) / n

Extensions

Added terms 19828..443892, Joerg Arndt, Oct 03 2012

a(12)-a(15) from Robin Visser, May 10 2024