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 $(\frac{V(n+1)}{V(n)}{)}_{n\ge 1}$ and $(V(n+1)-V(n){)}_{n\ge 1}$. We also prove that the sets $\{\frac{\psi (n)}{V(n)}:n\ge 1\}$ and $\{\frac{V(n)}{\varphi (n)}:n\ge 1\}$ are everywhere dense in $(1,+\infty )$.