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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.

%I A247105 #53 Jan 04 2025 14:07:17
%S A247105 1,5,25,109,385,1373,4645,16009,48817,159757,488377,1571425,4560901,
%T A247105 14482393,43408013,130394125,380755429,1118740741,3326930413,
%U A247105 9931863461,28466058257,84243573797,240453967777,706827067045,2009065808473,5913933615149,16711898903281
%N 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.
%C A247105 Starting with the natural numbers, make 2 passes removing every 2nd number, 3 passes removing every 3rd number, etc.
%C A247105 Is the limiting value of a(n+1)/a(n)=3?
%C A247105 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
%H A247105 Hiroaki Yamanouchi, <a href="/A247105/b247105.txt">Table of n, a(n) for n = 1..1000</a>
%H A247105 <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%e A247105 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.
%t A247105 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 *)
%o A247105 (PARI) copydropmult(v,m)=vector(#v-#v\m,i,v[(i-1)*m\(m-1)+1])
%o A247105 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
%o A247105 (Python)
%o A247105 for n in range(1, 101):
%o A247105   m = n
%o A247105   for i in range(n, 1, -1):
%o A247105     for j in range(i):
%o A247105       t = m * i // (i - 1)
%o A247105       while t - t // i >= m:
%o A247105         t -= 1
%o A247105       m = t + 1
%o A247105   print(f"{n} {m}") # _Hiroaki Yamanouchi_, Nov 28 2014
%Y A247105 Cf. A000960, A056533, A099204.
%K A247105 nonn
%O A247105 1,2
%A A247105 _Sergio Pimentel_, Nov 18 2014
%E A247105 More values from _Franklin T. Adams-Watters_, Nov 21 2014
%E A247105 a(12)-a(20) from _Alois P. Heinz_, Nov 26 2014
%E A247105 a(21)-a(27) from _Hiroaki Yamanouchi_, Nov 27 2014