A new upper bound for numbers with the Lehmer property and its application to repunit numbers

A composite positive integer $n$ has the Lehmer property if $\varphi (n)$ divides $n-1,$ where $\varphi$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\le 2^{2^K}-2^{2^{K-1}}$, where $K$ is the number of prime divisors of $n$. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set $$\{\frac{g^n-1}{g-1}\mid n,g\in \mathbb{N},{\nu }_2(g)+{\nu }_2(g+1)\le L\},$$ where ${\nu }_2(g)$ denotes the highest power of $2$ that divides $g$, and $L\ge 1$ is a fixed real number.