A114183 a(1) = 1; for n>1, a(n) = floor(sqrt(a(n-1))) if that number is not already in the sequence, otherwise a(n) = 2a(n-1).

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

Offset: 1

Author

Franklin T. Adams-Watters, Feb 04 2006

Keywords

• nonn
• look

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
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A189419 (inverse), A221715, A221716, A000196, A000523, A213218.

See A222193 and A222194 for records.

Programs

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

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 *)

Extensions

Missing negative in definition inserted by D. S. McNeil, May 26 2010

Entry revised by N. J. A. Sloane, Mar 01 2013