

A114183


a(1) = 1; for n>1, a(n) = floor(sqrt(a(n1))) if that number is not already in the sequence, otherwise a(n) = 2a(n1).


20



1, 2, 4, 8, 16, 32, 5, 10, 3, 6, 12, 24, 48, 96, 9, 18, 36, 72, 144, 288, 576, 1152, 33, 66, 132, 11, 22, 44, 88, 176, 13, 26, 52, 7, 14, 28, 56, 112, 224, 448, 21, 42, 84, 168, 336, 672, 25, 50, 100, 200, 400, 20, 40, 80, 160, 320, 17, 34, 68, 136, 272, 544, 23, 46, 92
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OFFSET

1,2


COMMENTS

One can prove by induction that n must appear in the sequence after [n/2], showing that the sequence is onetoone; 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. AdamsWatters, 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
Index entries for sequences that are permutations of the natural numbers


MAPLE

See A221715.


MATHEMATICA

a[1] = 1; a[n_] := a[n] = With[{an = Floor[Sqrt[a[n1]]]}, If[FreeQ[Array[a, n1], an], an, 2*a[n1]]]; Table[a[n], {n, 1, 65}] (* JeanFrançois Alcover, Apr 23 2013 *)


PROG

(Haskell)
a114183 n = a114183_list !! (n1)
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

Cf. A189419 (inverse), A221715, A221716, A000196, A000523, A213218.
See A222193 and A222194 for records.
Sequence in context: A318776 A036130 A122169 * A036129 A319303 A088976
Adjacent sequences: A114180 A114181 A114182 * A114184 A114185 A114186


KEYWORD

nonn,look


AUTHOR

Franklin T. AdamsWatters, 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



