A027380 Number of irreducible polynomials of degree n over GF(8); dimensions of free Lie algebras.

1, 8, 28, 168, 1008, 6552, 43596, 299592, 2096640, 14913024, 107370900, 780903144, 5726600880, 42288908760, 314146029564, 2345624803704, 17592184995840, 132458812569720, 1000799909722368, 7585009898729256

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of Lyndon words with 8 letters. - Joerg Arndt, Jul 29 2014

Number of aperiodic necklaces with n beads of 8 colors. - Herbert Kociemba, Nov 25 2016

Examples

			G.f. = 1 + 8*x + 28*x^2 + 168*x^3 + 1008*x^4 + 6552*x^5 + 43596*x^6 + ...
		

References

• E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
• M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1110 (terms 0..200 from T. D. Noe)
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.
• Index entries for sequences related to Lyndon words

Crossrefs

Column 8 of A074650.

Cf. A001037, A001693.

Programs

Maple

A027380 := proc(n)
    local d;
    if n = 0 then
        1;
    else
        add( 8^(n/d)*numtheory[mobius](d),d=numtheory[divisors](n)) ;
        %/n ;
    end if;
end proc: # R. J. Mathar, Jun 09 2016

Mathematica

f[n_] := (1/n)*Sum[MoebiusMu[d]*8^(n/d), {d, Divisors[n]}]; f[0] = 1; Array[f, 20, 0] (* Robert G. Wilson v, Jul 28 2014 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,8],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

PARI

a(n) = if(n, sumdiv(n, d, moebius(d)*8^(n/d))/n, 1) \\ Altug Alkan, Dec 01 2015

Formula

G.f.: k=8, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

a(n) = Sum_{d|n} mu(d)*8^(n/d)/n for n > 0. - Andrew Howroyd, Oct 13 2017