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A059991 a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))). 3


%S 1,1,4,1,16,16,64,1,256,256,1024,256,4096,4096,16384,1,65536,65536,

%T 262144,65536,1048576,1048576,4194304,65536,16777216,16777216,

%U 67108864,16777216,268435456,268435456,1073741824,1,4294967296,4294967296

%N a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))).

%C Number of points of period n in the simplest nontrivial disconnected S-integer dynamical system.

%C This sequence comes from the simplest disconnected S-integer system that is not hyperbolic. In the terminology of the papers referred to, it is constructed by choosing the under- lying field to be F_2(t), the element to be t and the nontrivial valuation to correspond to the polynomial 1+t. Since it counts periodic points, it satisfies the nontrivial congruence sum_{d|n}mu(d)a(n/d) = 0 mod n for all n and since it comes from a group automorphism it is a divisibility sequence.

%H R. Brown and J. L. Merzel, <a href="http://www.fq.math.ca/Papers1/45-2/brown.pdf">The number of Ducci sequences with a given period</a>, Fib. Quart., 45 (2007), 115-121.

%H Vijay Chothi, <a href="http://www.mth.uea.ac.uk/admissions/graduate/phds.html">Periodic Points in S-integer dynamical systems</a>, PhD thesis, University of East Anglia, 1996. [Broken link]

%H Vijay Chothi, Graham Everest, Thomas Ward, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN00221492X">S-integer dynamical systems: periodic points</a>, J. Reine Angew Math. 489 (1997), 99-132.

%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://nyjm.albany.edu/Conf/Ergodic/Ward.ps">Almost all S-integer dynamical systems have many periodic points</a>, Ergodic Th. Dynam. Sys. 18 (1998), 471-486.

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%e a(24) = 2^16 = 65536 because ord_2(24)=3, so 24-2^ord_2(24)=16.

%p readlib(ifactors): for n from 1 to 100 do if n mod 2 = 1 then ord2 := 0 else ord2 := ifactors(n)[2][1][2] fi: printf(`%d,`, 2^(n-2^ord2)) od:

%t ord2[n_?OddQ] = 0; ord2[n_?EvenQ] := FactorInteger[n][[1, 2]]; a[n_] := 2^(n-2^ord2[n]); a /@ Range[34]

%t (* _Jean-Fran├žois Alcover_, May 19 2011, after Maple prog. *)

%Y Cf. A000079, A006519, A129760.

%K easy,nonn

%O 1,3

%A Thomas Ward (t.ward(AT)uea.ac.uk), Mar 08 2001

%E More terms from _James A. Sellers_, Mar 15 2001

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)