Bulletin of the
Korean Mathematical Society BKMS

We say a positive integer $n$ satisfies the Lehmer property if $\phi(n)$ divides $n-1$, where $\phi(n)$ is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form $D_{p,n}=np^n+1$, for a prime $p$ and a positive integer $n$, or of the form $\alpha2^{\beta}+1$ for $\alpha \le \beta$ does not satisfy the Lehmer property.

Keywords: Euler's totient function, generalized Cullen number, Lehmer property

MSC numbers: 11A05, 11N25