A192509 Number of conjugacy classes of primitive elements in GF(7^n) which have trace 0.

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

Offset: 1

Author

Joerg Arndt, Jul 03 2011

Keywords

• nonn
• hard
• more

Comments

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

Crossrefs

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

Programs

GAP

p := 7;
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=7)

Formula

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

Extensions

a(7)-a(9) from Joerg Arndt, Oct 14 2012

a(10)-a(12) from Robin Visser, Jun 01 2024