%I
%S 1,1,1,2,2,4,8,16,16,32,64,128,256,512,1024,2048,2048,4096,8192,16384,
%T 32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,
%U 16777216,33554432,67108864,67108864,134217728,268435456,536870912
%N Number of partitions of n into distinct positive parts <= n, where parts are combined by XOR.
%C Usually successive powers of 2, but "stutters" when n is power of 2. (G.f. must satisfy some interesting functional equations!). Empty partition of 0 defined as 1.
%C This is an instance of what I like to call "numbral theory": whenever you have a set of indexed objects that you can do some kind of arithmetic on, then the indices act as "shadows" of the objects and you can generally talk about lots of analogs, such as partitions, primes, even generating functions, etc. It would be worthwhile to systematically "fill out" the entries for as many of these systems as possible in the OEIS.
%C The "AND" version is just the allones sequence.  _Christian G. Bower_, Jun 07 2005
%C a(n) is the number of orbits of the FlipAfter1 map on integers with n+1 binary digits. The FlipAfter1 map on an integer in binary form is: flip each bit that is immediately preceded by a "1". For example, the orbits on 4bit numbers are 1000 > 1100 > 1010 > 1111 and 1001 > 1101 > 1011 > 1110. The orbits on nbit numbers are all of length 2^floor(log_2(n1)+1) (for n >= 2), A062383. There is precisely one member of each orbit in the following set: integers in binary form such that each bit at distance a power of two from the leading "1" is 0. This set of orbit representatives begins 1, 10, 100, 1000, 1001, 10000, 10010, 100000, 100001, 100100, 100101.  _David Callan_, Oct 13 2012
%H T. D. Noe, <a href="/A054243/b054243.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = 2^floor(n  log_2(n)  1) = A000079(n)/A062383(n).  _Henry Bottomley_, Nov 22 2001
%e a(5)=4 thus: 5 4+1 5+3+2+1 4+3+2 (where "+" = XOR).
%t a[n_] := 2^(n  Ceiling[ Log[2, n] ]  1); Table[a[n], {n, 1, 36}] (* _JeanFrançois Alcover_, Apr 04 2013 *)
%o (PARI) a(n)=2^(n1log(n+.5)\log(2)) \\ _Charles R Greathouse IV_, Apr 08 2012
%Y InclusiveOR (or IOR) version: A054244.
%Y Cf. A160473.
%K easy,nonn,nice
%O 0,4
%A _Marc LeBrun_, Feb 08 2000
