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A060220 Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026).

%I A060220 #23 Jan 05 2025 19:51:36
%S A060220 17,136,1632,20808,283968,4022064,58619808,871959240,13176430176,
%T A060220 201599248032,3115626937056,48551851084080,761890617915840,
%U A060220 12026987582075856,190828203433892736,3041324491793194440,48661191875666868480,781282469552728498992,12582759772902701307744
%N A060220 Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026).
%C A060220 Number of monic irreducible polynomials of degree n over GF(17). - _Andrew Howroyd_, Dec 10 2017
%H A060220 G. C. Greubel, <a href="/A060220/b060220.txt">Table of n, a(n) for n = 1..810</a>
%H A060220 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 A060220 Yash Puri and Thomas Ward, <a href="https://web.archive.org/web/2024*/https://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 A060220 T. Ward, <a href="http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>
%F A060220 a(n) = (1/n)* Sum_{d|n} mu(d)*A001026(n/d).
%F A060220 G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 17*x^k))/k. - _Ilya Gutkovskiy_, May 20 2019
%e A060220 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 A060220[n_]:= DivisorSum[n, (17)^(n/#)*MoebiusMu[#] &]/n;
%t A060220 Table[A060220[n], {n,40}] (* _G. C. Greubel_, Sep 13 2024 *)
%o A060220 (PARI) a001024(n) = 17^n;
%o A060220 a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ _Michel Marcus_, Sep 11 2017
%o A060220 (Magma)
%o A060220 A060220:= func< n | (1/n)*(&+[MoebiusMu(d)*(17)^Floor(n/d): d in Divisors(n)]) >;
%o A060220 [A060220(n): n in [1..40]]; // _G. C. Greubel_, Sep 13 2024
%o A060220 (SageMath)
%o A060220 def A060220(n): return (1/n)*sum(moebius(k)*(17)^(n/k) for k in (1..n) if (k).divides(n))
%o A060220 [A060220(n) for n in range(1,41)] # _G. C. Greubel_, Sep 13 2024
%Y A060220 Column 17 of A074650.
%Y A060220 Cf. A001026, A008683.
%K A060220 easy,nonn
%O A060220 1,1
%A A060220 _Thomas Ward_, Mar 21 2001
%E A060220 More terms from _Michel Marcus_, Sep 11 2017