

A247105


Variation of Flavius Josephus's sieve: Start with the natural numbers; at the kth sieving step, make k passes removing every kth term of the sequence remaining after the previous sieving step; iterate.


1



1, 5, 25, 109, 385, 1373, 4645, 16009, 48817, 159757, 488377, 1571425, 4560901, 14482393, 43408013, 130394125, 380755429, 1118740741, 3326930413, 9931863461, 28466058257, 84243573797, 240453967777, 706827067045, 2009065808473, 5913933615149, 16711898903281
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OFFSET

1,2


COMMENTS

Starting with the natural numbers, make 2 passes removing every 2nd number, 3 passes removing every 3rd number, etc.
Is the limiting value of a(n+1)/a(n)=3?
Since 1/(11/n)^n converges to e (as n > inf), a(n+1)/a(n) converges to e.  Hiroaki Yamanouchi, Nov 27 2014


LINKS

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to the Josephus Problem


EXAMPLE

The 1st pass removes 2, 4, 6, 8, 10, etc. The 2nd pass (also with 2) removes 3, 7, 11, 15, 19, etc. Then there are 3 passes removing every 3rd number, of which the 1st pass removes 9, 21, 33, 45, ..., the 2nd removes 13, 29, 49, ..., and the 3rd removes 17, 41, 73, ...; then there are 4 passes with 4; 5 passes with 5; etc.


MATHEMATICA

A247105 = Reap[Quiet @ For[n=1, n<28, n++, m = n; For[i=n, i >= 1, i, For[j=1, j <= i, j++, t = Floor[(m*i)/(i1)]; While[t  Floor[t/i] >= m, t = 1]; om = m; m = t+1]]; Sow[om]]][[2, 1]] (* JeanFrançois Alcover, Nov 28 2014, translated and adapted from Hiroaki Yamanouchi's Python script *)


PROG

(PARI) copydropmult(v, m)=vector(#v#v\m, i, v[(i1)*m\(m1)+1])
alim(n)=my(r=vector(n, i, i), j=2, k=1); while(j<#r, r=copydropmult(r, j); if(k++>j, j++; k=1)); r
(Python)
for n in range(1, 101):
..m = n
..for i in range(n, 1, 1):
....for j in range(i):
......t = m * i // (i  1)
......while t  t // i >= m:
........t = 1
......m = t + 1
..print("%d %d" % (n, m)) # Hiroaki Yamanouchi, Nov 28 2014


CROSSREFS

Cf. A000960, A056533, A099204.
Sequence in context: A224129 A111641 A074419 * A272249 A275903 A273828
Adjacent sequences: A247102 A247103 A247104 * A247106 A247107 A247108


KEYWORD

nonn


AUTHOR

Sergio Pimentel, Nov 18 2014


EXTENSIONS

More values from Franklin T. AdamsWatters, Nov 21 2014
a(12)a(20) from Alois P. Heinz, Nov 26 2014
a(21)a(27) from Hiroaki Yamanouchi, Nov 27 2014


STATUS

approved



