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 di er 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.