A152423 A variation of the Josephus problem, removing every other person, starting with person 1; a(n) is the last person remaining.

1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 8, 10, 12, 14, 16, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

Offset

1,2

Comments

Begin with n people standing in a circle, numbered clockwise 1 through n. Until only one person remains, go around the circle clockwise, removing every other person, starting by removing person 1. a(n) is the number of the last person remaining.

Apparently a(n) = 2*A062050(n-1), n > 1. - Paul Curtz, May 30 2011

Links

Alois P. Heinz, Table of n, a(n) for n = 1..8192

Eric Weisstein's World of Mathematics, Josephus Problem

Wikipedia, Josephus problem

Index entries for sequences related to the Josephus Problem

Formula

a(1)=1, a(2)=2; for n > 2, a(n)=2 if n < a(n-1) + 2, otherwise a(n) = a(n-1) + 2.

a(n)=n if n is a power of 2, otherwise a(n)=2*(n-2^m) where m is the exponent of the nearest power of 2 below n. - Nicolas Patrois, Apr 19 2021

a(n) = 2*n - 2^ceiling(log_2(n)). - Alois P. Heinz, Nov 22 2023

Example

From Omar E. Pol, Dec 16 2013: (Start)
It appears that this is also an irregular triangle with row lengths A011782 as shown below:
1;
2;
2,4;
2,4,6,8;
2,4,6,8,10,12,14,16;
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32;
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40, 42,44,46,48,50,52,54,56,58,60,62,64;
Right border gives A000079.
(End)

Maple

a:= n-> 2*n - 2^ceil(log[2](n)):
seq(a(n), n=1..74);  # Alois P. Heinz, Nov 22 2023

Mathematica

A152423[n_]:=2n-2^Ceiling[Log2[n]]; Array[A152423, 100] (* Paolo Xausa, Nov 23 2023 *)

Prog

(PHP) function F($in){ $a[1] = 1; if($in == 1){ return $a; } $temp =2; for($i=2; $i<=$in; $i++){ $temp+=2; if($temp>$i){ $temp = 2 ; } $answer[] = $temp; } return $answer; } #change $n value for the result $n=5; #sequence store in $answer by using $a = F($n); #to display a(n) echo $a[n];
(Python) m=len(bin(n))-3; print(n if 2**m==n else 2*(n-2**m)) # Nicolas Patrois, Apr 19 2021

Crossrefs

The Index to the OEIS lists 21 entries under "Josephus problem". - N. J. A. Sloane, Dec 04 2008

Cf. A000079, A011782, A062050.

Sequence in context: A063789 A106264 A278535 * A233765 A233781 A233971

Adjacent sequences: A152420 A152421 A152422 * A152424 A152425 A152426

Keyword

easy,nonn

Author

Suttapong Wara-asawapati (retsam_krad(AT)hotmail.com), Dec 03 2008

Extensions

Edited by Jon E. Schoenfield, Feb 29 2020

Status

approved