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

1,2

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

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.13, p. 131.

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.

Example

1.62227050288476731595695...

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 *)

Prog

(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

Crossrefs

Cf. A054995, A083287.

Sequence in context: A020795 A136710 A276801 * A247818 A325039 A172439

Adjacent sequences: A083283 A083284 A083285 * A083287 A083288 A083289

Keyword

nonn,cons,changed

Author

Ralf Stephan, Apr 23 2003

Status

approved