Analytical Study and Efficient Evaluation of the Josephus Function

A new approach to analyzing intrinsic properties of the Josephus function, $J_{{}_k}$, is presented in this paper. The linear structure between extreme points of $J_{{}_k}$ is fully revealed, leading to the design of an efficient algorithm for evaluating $J_{{}_k}(n)$. Algebraic expressions that describe how recursively compute extreme points, including fixed points, are derived. The existence of consecutive extreme and also fixed points for all $k\ge 2$ is proven as a consequence, which generalizes Knuth result for $k=2$. Moreover, an extensive comparative numerical experiment is conducted to illustrate the performance of the proposed algorithm for evaluating the Josephus function compared to established algorithms. The results show that the proposed scheme is highly effective in computing $J_{{}_k}(n)$ for large inputs.