A105154 Consider trajectory of n under repeated application of map k -> A105027(k); a(n) = length of cycle.

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, 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, 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

Offset: 0

Author

Philippe Deléham, Apr 30 2005

Keywords

• nonn
• easy
• base

Comments

Why is this always a power of 2?

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.

Links

• Hagen von Eitzen, Table of n, a(n) for n = 0..10000
• David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Crossrefs

Cf. A102370, A105025, A105027, A105153.

Programs

Haskell

a105154 n = t [n] where
   t xs@(x:_) | y `elem` xs = length xs
              | otherwise   = t (y : xs) where y = a105027 x
-- Reinhard Zumkeller, Jul 21 2012

Extensions

More terms taken from b-file by Hagen von Eitzen, Jun 24 2009