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A083286
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Decimal expansion of K(3), a constant related to the Josephus problem.
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12
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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E. T. H. Wang, Phillip C. Washburn, Problem E2604, American Mathematical Monthly, 84 (1977), 821-822.
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EXAMPLE
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1.62227050288476731595695...
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MATHEMATICA
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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]
(* Display of the surroundings of 3/2 *)
Plot[N[c[x, 20]], {x, 1, 3}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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