OFFSET
1,2
COMMENTS
One can prove by induction that n must appear in the sequence after [n/2], showing that the sequence is one-to-one; and that frac(log_2(log_2(a(n))) is dense in [0,1), from which it follows that a(n) is onto. - From Franklin T. Adams-Watters, Feb 04 2006
Comment from N. J. A. Sloane, Mar 01 2013: Although the preceding argument seems somewhat incomplete, the result is certainly true: This sequence is a permutation of the natural numbers. Mark Hennings and the United Kingdom Mathematics Trust, and (independently) Max Alekseyev, sent detailed proofs - see the link below.
The sequence consists of a series of "doubling runs", and the starting points and lengths of these runs are in A221715 and A221716 respectively. - N. J. A. Sloane, Jan 27 2013
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Mark Hennings and the United Kingdom Mathematics Trust, Proof that the sequence is a permutation of the natural numbers. An essentially identical proof was contributed by Max Alekseyev.
N. J. A. Sloane, Table of n, a(n) for n = 1..100000
MAPLE
See A221715.
MATHEMATICA
a[1] = 1; a[n_] := a[n] = With[{an = Floor[Sqrt[a[n-1]]]}, If[FreeQ[Array[a, n-1], an], an, 2*a[n-1]]]; Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Apr 23 2013 *)
PROG
(Haskell)
a114183 n = a114183_list !! (n-1)
a114183_list = 1 : f [1] where
f xs@(x:_) = y : f (y : xs) where
y = if z `notElem` xs then z else 2 * x where z = a000196 x
-- Reinhard Zumkeller, Mar 05 2013
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Franklin T. Adams-Watters, Feb 04 2006
EXTENSIONS
Missing negative in definition inserted by D. S. McNeil, May 26 2010
Entry revised by N. J. A. Sloane, Mar 01 2013
STATUS
approved