A composite positive integer $n$ has the Lehmer property if $\phi(n)$ divides $n-1,$ where $\phi$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\leq 2^{ 2^{ K}}-2^{ 2^{ K-1}}$, 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 $$ \left\{ \frac{ g^{ n}-1}{ g-1}\ \bigg|\ n,g\in\mathbb{ N},\ \nu_{ 2}(g)+\nu_{ 2}(g+1)\leq L\ \right\}, $$ where $\nu_{ 2}(g)$ denotes the highest power of $2$ that divides $g$, and $L\geq 1$ is a fixed real number.
Categories: math.NT