The Josephus problem
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 ~ 4) 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.