A083286 - OEIS

%I #48 Jul 21 2020 14:37:31

%S 1,6,2,2,2,7,0,5,0,2,8,8,4,7,6,7,3,1,5,9,5,6,9,5,0,9,8,2,8,9,9,3,2,4,

%T 1,1,3,0,6,6,1,0,5,5,6,2,3,1,3,0,3,7,4,3,2,1,8,5,4,4,3,3,8,7,3,7,8,4,

%U 3,3,9,9,9,7,2,7,4,8,4,4,7,6,3,8,3,6,1,6,5,3,9,8,3,3,2,3,3,4,1,1,0,0

%N Decimal expansion of K(3), a constant related to the Josephus problem.

%C The constant K(3) is related to the Josephus problem with q=3 and the computation of A054995.

%C The number also occurs in Washburn's solution cited in References. Regarding Washburn's limit more generally (with x in place of 3/2) results in a disconnected function as plotted by the Mathematica program below. - _Clark Kimberling_, Oct 24 2012

%H A.H.M. Smeets, <a href="/A083286/b083286.txt">Table of n, a(n) for n = 1..20000</a>

%H A. M. Odlyzko and H. S. Wilf, <a href="https://doi.org/10.1017/S0017089500008272">Functional iteration and the Josephus problem</a>, Glasgow Math. J. 33 (1991), 235-240.

%H A.H.M. Smeets, <a href="/A083286/a083286_2.txt">100000 decimal digits</a>

%H E. T. H. Wang, Phillip C. Washburn, <a href="http://www.jstor.org/stable/2322068">Problem E2604</a>, American Mathematical Monthly, 84 (1977), 821-822.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerCeilings.html">Power Ceilings</a>.

%H <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>

%e 1.62227050288476731595695...

%t s[x_, 0] := 0; s[x_, n_] := Floor[x*s[x, n - 1]] + 1

%t c[x_, n_] := ((1/x)^n) s[x, n]

%t t = N[c[3/2, 800], 120]

%t RealDigits[t, 10]  (* A083286 *)

%t (* Display of the surroundings of 3/2 *)

%t Plot[N[c[x, 20]], {x, 1, 3}]

%t (* _Clark Kimberling_, Oct 24 2012 *)

%o (PARI) p=1; N=10^4; for(n=1, N, p=ceil(3/2*p)); c=(p/(3/2)^N)+0.

%o (Python)

%o d, a, n, nmax = 3, 0, 0, 150000

%o while n < nmax:

%o     n, a = n+1, (a*d)//(d-1)+1

%o nom, den, pos = a*(d-1)**n, d**n, 0

%o while pos < 20000:

%o     dig, nom, pos = nom//den, (nom%den)*10, pos+1

%o     print(pos,dig) # _A.H.M. Smeets_, Jul 05 2019

%Y Cf. A054995, A083287.

%K nonn,cons

%O 1,2

%A _Ralf Stephan_, Apr 23 2003