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A060219 Number of orbits of length n under the full 16-shift (whose periodic points are counted by A001025).

%I A060219 #29 Jan 05 2025 19:51:36
%S A060219 16,120,1360,16320,209712,2795480,38347920,536862720,7635496960,
%T A060219 109951057896,1599289640400,23456246655680,346430740566960,
%U A060219 5146970983535160,76861433640386288,1152921504338411520,17361641481138401520,262353693488939386880,3976729669784964390480
%N A060219 Number of orbits of length n under the full 16-shift (whose periodic points are counted by A001025).
%C A060219 Number of monic irreducible polynomials of degree n over GF(16). - _Robert Israel_, Jan 07 2015
%C A060219 Number of Lyndon words (aperiodic necklaces) with n beads of 16 colors. - _Andrew Howroyd_, Dec 10 2017
%H A060219 G. C. Greubel, <a href="/A060219/b060219.txt">Table of n, a(n) for n = 1..825</a>
%H A060219 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 A060219 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 A060219 T. Ward, <a href="http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>
%F A060219 a(n) = (1/n)* Sum_{d|n} mu(d)*16^(n/d).
%F A060219 G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 16*x^k))/k. - _Ilya Gutkovskiy_, May 19 2019
%e A060219 a(2)=120 since there are 256 points of period 2 in the full 16-shift and 16 fixed points, so there must be (256-16)/2 = 120 orbits of length 2.
%p A060219 f:= (n,p) -> add(numtheory:-mobius(d)*p^(n/d),d=numtheory:-divisors(n))/n:
%p A060219 seq(f(n,16),n=1..30); # _Robert Israel_, Jan 07 2015
%t A060219 A060219[n_]:= DivisorSum[n, MoebiusMu[#]*16^(n/#) &]/n;Table[A060219[n], {n, 40}] (* _G. C. Greubel_, Aug 01 2024 *)
%o A060219 (PARI) a(n) = sumdiv(n, d, moebius(d)*16^(n/d))/n; \\ _Michel Marcus_, Jan 07 2015
%o A060219 (Magma)A060219:= func< n | (&+[MoebiusMu(d)*16^Floor(n/d): d in Divisors(n)])/n >;
%o A060219 [A060219(n): n in [1..40]]; // _G. C. Greubel_, Aug 01 2024
%o A060219 (SageMath)
%o A060219 def A060219(n): return sum(moebius(k)*16^(n//k) for k in (1..n) if (k).divides(n))/n
%o A060219 [A060219(n) for n in range(1, 41)] # _G. C. Greubel_, Aug 01 2024
%Y A060219 Column 16 of A074650.
%Y A060219 Cf. A001025.
%K A060219 easy,nonn
%O A060219 1,1
%A A060219 _Thomas Ward_, Mar 21 2001
%E A060219 Terms a(17) onward added by _G. C. Greubel_, Aug 01 2024