A105154 - OEIS

%I #17 Mar 04 2018 17:01:37

%S 1,1,2,2,2,1,2,1,4,2,2,4,4,2,2,4,4,4,1,4,4,4,1,4,4,4,1,4,4,4,1,4,8,4,

%T 4,4,4,8,4,4,8,4,4,4,4,8,4,4,8,4,4,4,4,8,4,4,8,4,4,4,4,8,4,4,16,8,4,2,

%U 4,8,16,2,16,8,4,2,4,8,16,2,16,8,4,2,4,8,16,2,16,8,4,2,4,8,16,2,16,8,4,2,4

%N Consider trajectory of n under repeated application of map k -> A105027(k); a(n) = length of cycle.

%C Why is this always a power of 2?

%C a(n) is always a power of 2: If n is a k-bit number, then so are all numbers in the A105154-orbit of n. For m in the orbit, the i-th bit (i=1,..,k) of A105154(m) is the i-th bit of m-k+i and hence depends only on the lower i bits of m. By induction quickly follows that the lower i bits run through a cycle of length dividing 2^i. This also shows that a(n) <= n for n > 0.

%H Hagen von Eitzen, <a href="/A105154/b105154.txt">Table of n, a(n) for n = 0..10000</a>

%H David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [<a href="http://neilsloane.com/doc/slopey.pdf">pdf</a>, <a href="http://neilsloane.com/doc/slopey.ps">ps</a>].

%o (Haskell)

%o a105154 n = t [n] where

%o    t xs@(x:_) | y `elem` xs = length xs

%o               | otherwise   = t (y : xs) where y = a105027 x

%o -- _Reinhard Zumkeller_, Jul 21 2012

%Y Cf. A102370, A105025, A105027, A105153.

%K nonn,easy,base

%O 0,3

%A _Philippe Deléham_, Apr 30 2005

%E More terms taken from b-file by _Hagen von Eitzen_, Jun 24 2009