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A001693 Number of degree-n irreducible polynomials over GF(7); dimensions of free Lie algebras.

%I A001693 M4373 N1838 #52 Nov 22 2017 16:41:10
%S A001693 1,7,21,112,588,3360,19544,117648,720300,4483696,28245840,179756976,
%T A001693 1153430600,7453000800,48444446376,316504099520,2077057800300,
%U A001693 13684147881600,90467419857752,599941851861744
%N A001693 Number of degree-n irreducible polynomials over GF(7); dimensions of free Lie algebras.
%C A001693 Number of aperiodic necklaces with n beads of 7 colors. - _Herbert Kociemba_, Nov 25 2016
%D A001693 E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
%D A001693 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
%D A001693 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001693 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001693 Seiichi Manyama, <a href="/A001693/b001693.txt">Table of n, a(n) for n = 0..1186</a> (terms 0..200 from T. D. Noe)
%H A001693 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A001693 G. J. Simmons, <a href="http://www.jstor.org/stable/2316211">The number of irreducible polynomials of degree n over GF(p)</a>, Amer. Math. Monthly, 77 (1970), 743-745.
%H A001693 G. Viennot, <a href="http://dx.doi.org/10.1007/BFb0067950">Algèbres de Lie Libres et Monoïdes Libres</a>, Lecture Notes in Mathematics 691, Springer Verlag 1978.
%H A001693 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A001693 a(n) = (1/n)*Sum_{d|n} mu(d)*7^(n/d), for n>0.
%F A001693 G.f.: k=7, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - _Herbert Kociemba_, Nov 25 2016
%p A001693 with(numtheory); A001693 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*7^(n/d); od; RETURN(s/n); fi; end;
%t A001693 a[n_]:=(1/n)*Sum[MoebiusMu[d]*7^(n/d), {d, Divisors[n]}]; a[0] = 1; Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Aug 31 2011, after formula *)
%t A001693 mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,7],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 25 2016 *)
%o A001693 (PARI) a(n) = if(n, sumdiv(n, d, moebius(d)*7^(n/d))/n, 1) \\ _Altug Alkan_, Dec 01 2015
%Y A001693 Column 7 of A074650.
%Y A001693 Cf. A027376, A000031, A001037, A032164.
%K A001693 nonn,easy,nice
%O A001693 0,2
%A A001693 _N. J. A. Sloane_
%E A001693 Description corrected by _Vladeta Jovovic_, Feb 09 2001