  In this paper, we introduce and study the iterates of the following family of
functions $\varphi_k$ defined on natural numbers which exhibits nice
properties. $$\varphi_k(x)=\left\lbrace \begin{array}{ll} x+k, & \mbox{ if $x$
is prime;}\\ \mbox{largest prime divisor of $x$,} & \mbox{ if $x$ is
composite;} \end{array} \right.$$ In particular, we study the periodic
behaviour of the trajectories of these iterated functions. In some cases, we
provide proofs of these properties and in some other cases we pose some open
problems based on numerical evidences supported by heuristic arguments.
