abstract
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The Josephus problem

Lorenz Halbeisen and Norbert Hungerbühler

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We give explicit non-recursive formulas to compute the
Josephus-numbers *j*(n,2,i) and *j*(n,3,i) and explicit upper and
lower bounds for *j*(n,k,i) (where k > 3) which differ by 2k-2
(for k=4 the bounds are even better). Furthermore we present a new
fast algorithm to calculate *j*(n,k,i) which is based upon the
mentioned bounds.