This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A059991 #25 Jul 02 2025 16:02:01
%S A059991 1,1,4,1,16,16,64,1,256,256,1024,256,4096,4096,16384,1,65536,65536,
%T A059991 262144,65536,1048576,1048576,4194304,65536,16777216,16777216,
%U A059991 67108864,16777216,268435456,268435456,1073741824,1,4294967296,4294967296
%N A059991 a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))).
%C A059991 Number of points of period n in the simplest nontrivial disconnected S-integer dynamical system.
%C A059991 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 A059991 R. Brown and J. L. Merzel, <a href="https://web.archive.org/web/2024*/https://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 A059991 Vijay Chothi, <a href="https://www.math.stonybrook.edu/theses/thesis96-1/part1.pdf">Periodic Points in S-integer dynamical systems</a>, PhD thesis, University of East Anglia, 1996.
%H A059991 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 A059991 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 A059991 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 A059991 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%e A059991 a(24) = 2^16 = 65536 because ord_2(24)=3, so 24-2^ord_2(24)=16.
%p A059991 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 A059991 ord2[n_?OddQ] = 0; ord2[n_?EvenQ] := FactorInteger[n][[1, 2]]; a[n_] := 2^(n-2^ord2[n]); a /@ Range[34]
%t A059991 (* _Jean-François Alcover_, May 19 2011, after Maple prog. *)
%Y A059991 Cf. A000079, A006519, A129760.
%K A059991 easy,nonn
%O A059991 1,3
%A A059991 _Thomas Ward_, Mar 08 2001
%E A059991 More terms from _James Sellers_, Mar 15 2001