Some properties of regular integers modulo n

Abstract

Denoting by V(n) the number of regular integers $(\text{mod}n)$, we study properties of the sequences $\displaystyle \biggl(\frac{V(n+1)}{V(n)}\biggr)_{n\geq1}$ and $\displaystyle (V(n+1)-V(n))_{n\geq1}$. We also prove that the sets $\displaystyle\biggl \lbrace\frac{\psi(n)}{V(n)} : n\geq1\biggr \rbrace$ and $\displaystyle\biggl \lbrace\frac{V(n)}{\phi(n)} : n\geq1\biggr \rbrace$ are everywhere dense in $(1,+\infty)$.