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%I M3804 N1554 #77 Nov 17 2021 07:18:03
%S 1,5,10,40,150,624,2580,11160,48750,217000,976248,4438920,20343700,
%T 93900240,435959820,2034504992,9536718750,44878791360,211927516500,
%U 1003867701480,4768371093720,22706531339280
%N Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.
%C Exponents in expansion of Hardy-Littlewood constant C_5 = 0.409874885.. = A269843 as a product_{n>=2} zeta(n)^(-a(n)).
%C Number of aperiodic necklaces with n beads of 5 colors. - _Herbert Kociemba_, Nov 25 2016
%D E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003
%D M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A001692/b001692.txt">Table of n, a(n) for n = 0..1435</a> (terms 0..200 from T. D. Noe)
%H Alin Bostan, Alexander Marynych, and Kilian Raschel, <a href="https://arxiv.org/abs/1901.03002">On the least common multiple of several random~integers</a>, arXiv:1901.03002 [math.PR], 2019.
%H Jeremie Detrey, P. J. Spaenlehauer, and P. Zimmermann, <a href="https://members.loria.fr/PZimmermann/papers/rho.pdf">Computing the rho constant</a>, Preprint 2016.
%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H Gareth A. Jones and Alexander K. Zvonkin, <a href="https://www.labri.fr/perso/zvonkin/Research/ProjPrimesShort.pdf">Groups of prime degree and the Bateman-Horn conjecture</a>, 2021.
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]
%H 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 Dionisel Y. Regalado and Rodel Azura, <a href="https://rmrj.usjr.edu.ph/index.php/RMRJ/article/view/532">An Analytic Approximation to the Density of Twin Primes</a>, Recoletos Multidisciplinary Research Journal (2019) Vol. 6, No. 2.
%H 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 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 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F a(n) = Sum_{d|n} mu(d)*5^(n/d)/n, for n>0.
%F G.f.: k=5, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - _Herbert Kociemba_, Nov 25 2016
%t 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 *)
%t 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 *)
%o (PARI) a(n)=if(n,sumdiv(n,d,moebius(d)*5^(n/d))/n,1) \\ _Charles R Greathouse IV_, Jun 15 2011
%o (Haskell)
%o a001692 n = flip div n $ sum $
%o zipWith (*) (map a008683 divs) (map a000351 $ reverse divs)
%o where divs = a027750_row n
%o -- _Reinhard Zumkeller_, Oct 07 2015
%Y Cf. A001037, A054720, A002105.
%Y 5th column of A074650. - _Alois P. Heinz_, Aug 08 2008
%Y Cf. A008683, A000351, A027750.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_
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