A060220 - OEIS

%I #21 Sep 13 2024 08:01:01

%S 17,136,1632,20808,283968,4022064,58619808,871959240,13176430176,

%T 201599248032,3115626937056,48551851084080,761890617915840,

%U 12026987582075856,190828203433892736,3041324491793194440,48661191875666868480,781282469552728498992,12582759772902701307744

%N Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026).

%C Number of monic irreducible polynomials of degree n over GF(17). - _Andrew Howroyd_, Dec 10 2017

%H G. C. Greubel, <a href="/A060220/b060220.txt">Table of n, a(n) for n = 1..810</a>

%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 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 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)*A001026(n/d).

%F G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 17*x^k))/k. - _Ilya Gutkovskiy_, May 20 2019

%e a(2)=136 since there are 289 points of period 2 in the full 17-shift and 17 fixed points, so there must be (289-17)/2 = 136 orbits of length 2.

%t A060220[n_]:= DivisorSum[n, (17)^(n/#)*MoebiusMu[#] &]/n;

%t Table[A060220[n], {n,40}] (* _G. C. Greubel_, Sep 13 2024 *)

%o (PARI) a001024(n) = 17^n;

%o a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ _Michel Marcus_, Sep 11 2017

%o (Magma)

%o A060220:= func< n | (1/n)*(&+[MoebiusMu(d)*(17)^Floor(n/d): d in Divisors(n)]) >;

%o [A060220(n): n in [1..40]]; // _G. C. Greubel_, Sep 13 2024

%o (SageMath)

%o def A060220(n): return (1/n)*sum(moebius(k)*(17)^(n/k) for k in (1..n) if (k).divides(n))

%o [A060220(n) for n in range(1,41)] # _G. C. Greubel_, Sep 13 2024

%Y Column 17 of A074650.

%Y Cf. A001026, A008683.

%K easy,nonn

%O 1,1

%A _Thomas Ward_, Mar 21 2001

%E More terms from _Michel Marcus_, Sep 11 2017