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

1,5

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).

Links

Table of n, a(n) for n=1..35.

Formula

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

Example

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.

Prog

(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;

Crossrefs

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

Sequence in context: A287768 A197831 A244995 * A246788 A365278 A342802

Adjacent sequences: A152046 A152047 A152048 * A152050 A152051 A152052

Keyword

nonn,hard,more

Author

David A. Madore, Nov 21 2008

Extensions

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

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

Status

approved