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A027380
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Number of irreducible polynomials of degree n over GF(8); dimensions of free Lie algebras.
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4
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1, 8, 28, 168, 1008, 6552, 43596, 299592, 2096640, 14913024, 107370900, 780903144, 5726600880, 42288908760, 314146029564, 2345624803704, 17592184995840, 132458812569720, 1000799909722368, 7585009898729256
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
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LINKS
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FORMULA
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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
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EXAMPLE
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G.f. = 1 + 8*x + 28*x^2 + 168*x^3 + 1008*x^4 + 6552*x^5 + 43596*x^6 + ...
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MAPLE
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local d;
if n = 0 then
1;
else
add( 8^(n/d)*numtheory[mobius](d), d=numtheory[divisors](n)) ;
%/n ;
end if;
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = if(n, sumdiv(n, d, moebius(d)*8^(n/d))/n, 1) \\ Altug Alkan, Dec 01 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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