A083286 Decimal expansion of K(3), a constant related to the Josephus problem.

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, 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, 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

Offset: 1

Author

Ralf Stephan, Apr 23 2003

Keywords

• nonn
• cons

Comments

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

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

Examples

			1.62227050288476731595695...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.13 and 2.30.1, pp. 131, 196.

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..20000
• A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33 (1991), 235-240.
• A.H.M. Smeets, 100000 decimal digits.
• E. T. H. Wang, Phillip C. Washburn, Problem E2604, American Mathematical Monthly, 84 (1977), 821-822.
• Eric Weisstein's World of Mathematics, Power Ceilings.
• Index entries for sequences related to the Josephus Problem.

Crossrefs

Cf. A054995, A083287.

Programs

Mathematica

s[x_, 0] := 0; s[x_, n_] := Floor[x*s[x, n - 1]] + 1
c[x_, n_] := ((1/x)^n) s[x, n]
t = N[c[3/2, 800], 120]
RealDigits[t, 10]  (* A083286 *)
(* Display of the surroundings of 3/2 *)
Plot[N[c[x, 20]], {x, 1, 3}]
(* Clark Kimberling, Oct 24 2012 *)

PARI

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

Python

d, a, n, nmax = 3, 0, 0, 150000
while n < nmax:
    n, a = n+1, (a*d)//(d-1)+1
nom, den, pos = a*(d-1)**n, d**n, 0
while pos < 20000:
    dig, nom, pos = nom//den, (nom%den)*10, pos+1
    print(pos,dig) # A.H.M. Smeets, Jul 05 2019