A001693 Number of degree-n irreducible polynomials over GF(7); dimensions of free Lie algebras.
1, 7, 21, 112, 588, 3360, 19544, 117648, 720300, 4483696, 28245840, 179756976, 1153430600, 7453000800, 48444446376, 316504099520, 2077057800300, 13684147881600, 90467419857752, 599941851861744
Offset: 0
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.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1186 (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. J. Simmons, The number of irreducible polynomials of degree n over GF(p), Amer. Math. Monthly, 77 (1970), 743-745.
- 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
Programs
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Maple
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;
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Mathematica
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 *) 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 *) -
PARI
a(n) = if(n, sumdiv(n, d, moebius(d)*7^(n/d))/n, 1) \\ Altug Alkan, Dec 01 2015
Formula
a(n) = (1/n)*Sum_{d|n} mu(d)*7^(n/d), for n>0.
G.f.: k=7, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
Extensions
Description corrected by Vladeta Jovovic, Feb 09 2001
Comments