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A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.
(Formerly M3804 N1554)
66
1, 5, 10, 40, 150, 624, 2580, 11160, 48750, 217000, 976248, 4438920, 20343700, 93900240, 435959820, 2034504992, 9536718750, 44878791360, 211927516500, 1003867701480, 4768371093720, 22706531339280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Exponents in expansion of Hardy-Littlewood constant C_5 = 0.409874885.. as a product_{n>=2} zeta(n)^(-a(n)).

Number of aperiodic necklaces with n beads of 5 colors. - Herbert Kociemba, Nov 25 2016

REFERENCES

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.

Jeremie Detrey, PJ Spaenlehauer, P Zimmermann, Computing the rho constant, Preprint 2016, https://members.loria.fr/PZimmermann/papers/rho.pdf

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..1435 (terms 0..200 from T. D. Noe)

Steven R. Finch, Mathematical Constants

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

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) = Sum_{d|n} mu(d)*5^(n/d)/n, for n>0.

G.f.: k=5, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

MATHEMATICA

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*5^(n/d)/n, {d, Divisors[n]}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Mar 11 2014 *)

mx=40; f[x_, k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i, {i, 1, mx}]; CoefficientList[Series[f[x, 5], {x, 0, mx}], x] (* Herbert Kociemba, Nov 25 2016 *)

PROG

(PARI) a(n)=if(n, sumdiv(n, d, moebius(d)*5^(n/d))/n, 1) \\ Charles R Greathouse IV, Jun 15 2011

(Haskell)

a001692 n = flip div n $ sum $

            zipWith (*) (map a008683 divs) (map a000351 $ reverse divs)

            where divs = a027750_row n

-- Reinhard Zumkeller, Oct 07 2015

CROSSREFS

Cf. A001037, A054720, A002105.

5th column of A074650. - Alois P. Heinz, Aug 08 2008

Cf. A008683, A000351, A027750.

Sequence in context: A117865 A249059 A163305 * A038070 A136138 A270288

Adjacent sequences:  A001689 A001690 A001691 * A001693 A001694 A001695

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 17 22:48 EDT 2018. Contains 316297 sequences. (Running on oeis4.)