Some properties of the pseudo-Smarandache function

Abstract

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