A221716 -id:A221716 - cp's OEIS frontend

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

Franklin T. Adams-Watters, Feb 04 2006

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

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

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

See A222193 and A222194 for records.

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.
```

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 *)

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

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

A221715 Start of n-th doubling run in A114183.

Original entry on oeis.org

1, 5, 3, 9, 33, 11, 13, 7, 21, 25, 20, 17, 23, 19, 49, 39, 35, 47, 27, 29, 15, 43, 37, 97, 55, 41, 51, 57, 85, 73, 193, 111, 59, 61, 31, 89, 53, 329, 145, 192, 221, 237, 87, 105, 81, 101, 113, 481, 175, 149, 69, 93, 77, 99, 79, 71, 67, 65, 45, 75, 195, 157, 141, 189, 109, 83, 103, 229, 121, 352, 849, 233, 345, 297, 137, 187, 309, 281, 379, 155, 199, 159

Offset: 1

N. J. A. Sloane, Jan 27 2013

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..7328

Cf. A114183, A189419, A221716, A213220.

# A114181
M:=10000; M2:=1000;
s1:={1}; v0:=[1]; v1:=[1]; v2:=[]; vi:=Array(1..M2);
t1:=1; r1:=1; vi[1]:=1;
for n from 2 to M do
t2:=floor(sqrt(t1));
if t2 in s1 then
   v0:=[op(v0),2*t1]; s1:={op(s1),2*t1}; r1:=r1+1; t1:=2*t1;
   if t1<=M2 then vi[t1]:=n; fi;
else
   v0:=[op(v0),t2]; s1:={op(s1),t2}; v1:=[op(v1),t2]; v2:=[op(v2),r1]; r1:=1; t1:=t2;
   if t1<=M2 then vi[t1]:=n; fi;
fi;
od:
# A114183:
[seq(v0[i],i=1..nops(v0))];
# A221715:
[seq(v1[i],i=1..nops(v1))];
# A221716:
[seq(v2[i],i=1..nops(v2))];
# A189419:
[seq(vi[i],i=1..M2)];

A213220 Positions in A114183 where doubling runs begin.

Original entry on oeis.org

1, 7, 9, 15, 23, 26, 31, 34, 41, 47, 52, 57, 63, 68, 76, 82, 88, 95, 100, 106, 110, 118, 124, 133, 139, 145, 152, 159, 167, 174, 184, 191, 197, 204, 209, 218, 224, 236, 243, 252, 261, 270, 276, 284, 291, 299, 307, 319, 326, 334, 340, 348, 355, 363, 370, 377, 384, 391, 397, 405, 415, 423, 431, 440, 447, 454, 462, 472

Offset: 1

N. J. A. Sloane, Mar 02 2013

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..7328

Cf. A114183, A221715, A221716 (first differences).

A213656 Value of A114183 at end of n-th doubling run.

Original entry on oeis.org

32, 10, 96, 1152, 132, 176, 52, 448, 672, 400, 320, 544, 368, 2432, 1568, 1248, 2240, 752, 864, 232, 1920, 1376, 9472, 3104, 1760, 2624, 3264, 7296, 5440, 37376, 12352, 3552, 3776, 976, 7936, 2848, 108544, 21056, 37120, 49152, 56576, 7584, 11136, 6720, 10368, 12928, 231424, 30784, 22400, 4768, 8832, 5952, 9856

Offset: 1

N. J. A. Sloane, Mar 03 2013

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..7327

Cf. A114183, A221715, A221716, A213220.

A222802 When A114183 decreases in value for the n-th time, dropping to k (say), a(n) is the number of steps earlier that floor(k/2) appeared in A114183.

Original entry on oeis.org

5, 8, 12, 18, 19, 21, 25, 33, 36, 44, 53, 37, 53, 64, 14, 31, 32, 69, 71, 76, 77, 108, 120, 39, 93, 105, 123, 125, 157, 170, 52, 91, 93, 99, 190, 192, 89, 225, 238, 121, 72, 158, 251, 238, 251, 270, 205, 50, 209, 282, 284, 286, 287, 288, 289, 361, 385, 370, 281, 282, 340, 342, 344, 346, 309, 310, 312, 367, 460, 275

Offset: 1

N. J. A. Sloane, Mar 08 2013

nonn

The fact that, when a number k occurs in A114183, floor(k/2) has already appeared, is a key step in the proof that A114183 is a permutation of the natural numbers. This fact is obvious if k is the result of a doubling step. The present sequence is an attempt to gain insight into why it is true when k occurs at a square root step.

			The first 50 terms of A114183 are:
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.
The sequence decreases from 32 to 5, from 10 to 3, from 96 to 9, and so on.
The values of k are therefore 5, 3, 9, 33, 11, 13, 7, 21, 25, ...
and the corresponding values of floor(k/2) are 2, 1, 4, 16, 5, 6, 3, 10, 12, ...
Since 2 appeared in A114183 5 steps before 5, a(1) = 5,
since 1 appeared 8 steps before 3, a(2) = 8,
since 4 appeared 12 steps before 9, a(3) = 12, and so on.

N. J. A. Sloane, Table of n, a(n) for n = 1..7322

Cf. A114183, A221715, A221716, A213220.

cp's OEIS Frontend

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).

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A221715 Start of n-th doubling run in A114183.

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A213220 Positions in A114183 where doubling runs begin.

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A213656 Value of A114183 at end of n-th doubling run.

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A222802 When A114183 decreases in value for the n-th time, dropping to k (say), a(n) is the number of steps earlier that floor(k/2) appeared in A114183.

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