A247105 Variation of Flavius Josephus's sieve: Start with the natural numbers; at the k-th sieving step, make k passes removing every k-th term of the sequence remaining after the previous sieving step; iterate.

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

Offset: 1

Author

Sergio Pimentel, Nov 18 2014

Keywords

• nonn

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/(1-1/n)^n converges to e (as n -> inf), a(n+1)/a(n) converges to e. - Hiroaki Yamanouchi, Nov 27 2014

Examples

			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.
		

Links

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

Crossrefs

Cf. A000960, A056533, A099204.

Programs

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)/(i-1)]; While[t - Floor[t/i] >= m, t -= 1];  om = m; m = t+1]]; Sow[om]]][[2, 1]] (* Jean-François Alcover, Nov 28 2014, translated and adapted from Hiroaki Yamanouchi's Python script *)

PARI

copydropmult(v,m)=vector(#v-#v\m,i,v[(i-1)*m\(m-1)+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(f"{n} {m}") # Hiroaki Yamanouchi, Nov 28 2014

Extensions

More values from Franklin T. Adams-Watters, Nov 21 2014

a(12)-a(20) from Alois P. Heinz, Nov 26 2014

a(21)-a(27) from Hiroaki Yamanouchi, Nov 27 2014