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%I #29 May 08 2021 23:04:58
%S 13,78,728,7098,74256,804076,8964072,101962770,1178277464,13785812040,
%T 162923672184,1941506688940,23298085122480,281241165925044,
%U 3412392867581152,41588538022965570
%N Number of orbits of length n under the full 13-shift (whose periodic points are counted by A001022).
%C Number of monic irreducible polynomials of degree n over GF(13). - _Robert Israel_, Jan 07 2015
%C Number of Lyndon words (aperiodic necklaces) with n beads of 13 colors. - _Andrew Howroyd_, Dec 10 2017
%H Indranil Ghosh, <a href="/A060216/b060216.txt">Table of n, a(n) for n = 1..100</a>
%H Yash Puri and Thomas Ward, <a href="http://www.fq.math.ca/Scanned/39-5/puri.pdf">A dynamical property unique to the Lucas sequence</a>, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
%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 T. Ward, <a href="http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>
%F a(n) = (1/n)* Sum_{d|n} mu(d) 13^(n/d).
%F G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 13*x^k))/k. - _Ilya Gutkovskiy_, May 19 2019
%e a(2)=78 since there are 169 points of period 2 in the full 13-shift and 13 fixed points, so there must be (169-13)/2 = 78 orbits of length 2.
%p f:= n -> add(numtheory:-mobius(d)*13^(n/d),d=numtheory:-divisors(n))/n;
%p seq(f(n), n=1..100); # _Robert Israel_, Jan 07 2015
%t a[n_]:=(1/n) * Sum[MoebiusMu[d] *13^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 20}] (* _Indranil Ghosh_, Mar 26 2017 *)
%o (PARI) a(n) = sumdiv(n, d, moebius(d)*13^(n/d))/n; \\ _Michel Marcus_, Jan 07 2015
%o (Python)
%o from sympy import divisors, mobius
%o print([sum(mobius(d) * 13**(n//d) for d in divisors(n))//n for n in range(1, 21)]) # _Indranil Ghosh_, Mar 26 2017
%Y Column 13 of A074650.
%Y Cf. A001022.
%K nonn
%O 1,1
%A _Thomas Ward_, Mar 21 2001
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