Regular integers modulo n

Let $n=p_1^{{\nu }_1}\mathrm{.}\mathrm{.}\mathrm{.}{p}_r^{{\nu }_r}>1$ be an integer. An integer $a$ is called regular (mod $n$) if there is an integer $x$ such that $a^{2}x\equiv a$ (mod $n$). Let $\varrho (n)$ denote the number of regular integers $a$ (mod $n$) such that $1\le a\le n$. Here $\varrho(n)=(\phi(p_1^{\nu_1})+1)… (\phi(p_r^{\nu_r})+1)$,where$\phi(n)$ is the Euler function. In this paper we first summarize some basic properties of regular integers (mod $n$). Then in order to compare the rates of growth of the functions $\varrho (n)$ and $\varphi (n)$ we investigate the average orders and the extremal orders of the functions $\varrho (n)\mathrm{/}\varphi (n)$, $\varphi (n)\mathrm{/}\varrho (n)$ and $1\mathrm{/}\varrho (n)$.