A001693 Number of degree-n irreducible polynomials over GF(7); dimensions of free Lie algebras. (Formerly M4373 N1838)

1, 7, 21, 112, 588, 3360, 19544, 117648, 720300, 4483696, 28245840, 179756976, 1153430600, 7453000800, 48444446376, 316504099520, 2077057800300, 13684147881600, 90467419857752, 599941851861744

Offset

0,2

Comments

Number of aperiodic necklaces with n beads of 7 colors. - Herbert Kociemba, Nov 25 2016

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

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

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;

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 *)

Prog

(PARI) a(n) = if(n, sumdiv(n, d, moebius(d)*7^(n/d))/n, 1) \\ Altug Alkan, Dec 01 2015

Crossrefs

Column 7 of A074650.

Cf. A027376, A000031, A001037, A032164.

Sequence in context: A219152 A038184 A001185 * A321521 A061961 A028248

Adjacent sequences: A001690 A001691 A001692 * A001694 A001695 A001696

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Description corrected by Vladeta Jovovic, Feb 09 2001

Status

approved