This site is supported by donations to The OEIS Foundation.

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.