Some properties of the pseudo-Smarandache function

We answer a number of questions relating to the pseudo-Smarandache function Z(n). We show that the ratio of consecutive values $Z(n+1)\mathrm{/}Z(n)$ and $Z(n-1)\mathrm{/}Z(n)$ are unbounded; that $Z(2n)\mathrm{/}Z(n)$ is unbounded; that $n\mathrm{/}Z(n)$ takes every integer value infinitely often; and that the series $\sum_n 1/Z(n)^\alpha$isconvergentforany$\alpha > 1$.