A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras. (Formerly M3804 N1554)

1, 5, 10, 40, 150, 624, 2580, 11160, 48750, 217000, 976248, 4438920, 20343700, 93900240, 435959820, 2034504992, 9536718750, 44878791360, 211927516500, 1003867701480, 4768371093720, 22706531339280

Offset

0,2

Comments

Exponents in expansion of Hardy-Littlewood constant C_5 = 0.409874885.. = A269843 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.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003

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)

Alin Bostan, Alexander Marynych, and Kilian Raschel, On the least common multiple of several random~integers, arXiv:1901.03002 [math.PR], 2019.

Jeremie Detrey, P. J. Spaenlehauer, and P. Zimmermann, Computing the rho constant, Preprint 2016.

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

Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.

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.

Dionisel Y. Regalado and Rodel Azura, An Analytic Approximation to the Density of Twin Primes, Recoletos Multidisciplinary Research Journal (2019) Vol. 6, No. 2.

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 A364738 A136138

Adjacent sequences: A001689 A001690 A001691 * A001693 A001694 A001695

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved