A152049 Number of conjugacy classes of primitive elements in GF(2^n) which have trace 0.

0, 0, 1, 1, 3, 2, 9, 9, 23, 29, 89, 72, 315, 375, 899, 1031, 3855, 3886, 13797, 12000, 42328, 59989, 178529, 138256, 647969, 859841, 2101143, 2370917, 9204061, 8911060, 34636833, 33556537, 105508927, 168423669, 464635937

Offset: 1

Author

David A. Madore, Nov 21 2008

Keywords

• nonn
• hard
• more

Comments

Also number of primitive polynomials of degree n over GF(2) whose second-highest coefficient is 0.

Always less than A011260 (and exactly one half of it when 2^n-1 is prime).

Examples

			a(3)=1 because of the two primitive degree 3 polynomials over GF(2), namely t^3+t+1 and t^3+t^2+1, only the former has a zero next-to-highest coefficient.
Similarly, a(13)=315, because of half (4096) of the 8192 elements of GF(2^13) have trace 0 and all except 0 (since 1 has trace 1) are primitive, so there are 4095/13=315 conjugacy classes of primitive elements of trace 0.
		

Crossrefs

Cf. A192507 (GF(3^n)), A192508 (GF(5^n)), A192509 (GF(7^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).

Programs

GAP

a := function(n)
    local q,k,cnt,x; q:=2^n; k:=GF(2,n); cnt:=0;
    for x in k do
        if Trace(k, GF(2), x)=0*Z(2) and Order(x)=q-1 then
            cnt := cnt+1;
        fi;
    od;
    return cnt/n;
end;
for n in [1..32] do  Print (a(n), ", ");  od;

Formula

a(n) = A192211(n)/n. [Joerg Arndt, Jul 03 2011]

Extensions

More terms (13797...8911060) by Joerg Arndt, Jun 26 2011.

More terms (34636833...464635937) by Joerg Arndt, Jul 03 2011.