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A192507 Number of conjugacy classes of primitive elements in GF(3^n) which have trace 0. 8
0, 0, 1, 2, 7, 14, 52, 104, 333, 870, 2571, 4590, 20440, 56736, 133782, 327558, 1265391 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also number of primitive polynomials of degree n over GF(3) whose second-highest coefficient is 0.

LINKS

Table of n, a(n) for n=1..17.

FORMULA

a(n) = A192212(n) / n.

PROG

(GAP)

p := 3;

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) # much more efficient

p=3; # choose characteristic

for n in xrange(1, 66):

    F=GF(p^n, 'x');

    g = F.multiplicative_generator(); # generator

    vt = vector(ZZ, p); # stats: trace

    m = p^n - 1; # size of multiplicative group

    ## Compute all irreducible polynomials via Lyndon words:

    for w in LyndonWords(p, n): # digits of Lyndon words range form 1, .., p

        e = sum( (w[j]-1) * p^j for j in xrange(0, n) )

        if gcd(m, e) == 1: # primitive elements only

            f = g^e;

            t = f.trace().lift(); # trace (over ZZ)

            vt[t] += 1;

    print vt[0]; # choose index 0, 1, .., p-1 for different traces

# Joerg Arndt, Oct 03 2012

CROSSREFS

Cf. A152049 (GF(2^n)), A192507 (GF(5^n)), A192509 (GF(7^n)), A192510 (GF(11^n)), A192511 (GF(13^n)).

Cf. A027385 (number of primitive polynomials of degree n over GF(3)).

Sequence in context: A018281 A018592 A123752 * A018622 A018668 A173756

Adjacent sequences:  A192504 A192505 A192506 * A192508 A192509 A192510

KEYWORD

nonn,hard,more

AUTHOR

Joerg Arndt, Jul 03 2011

EXTENSIONS

Added terms >=2571, Joerg Arndt, Oct 03 2012

STATUS

approved

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Last modified December 9 23:16 EST 2016. Contains 278993 sequences.