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 Generalized Cullen numbers with the Lehmer property Bull. Korean Math. Soc. 2013 Vol. 50, No. 6, 1981-1988 https://doi.org/10.4134/BKMS.2013.50.6.1981Published online November 1, 2013 Dae-June Kim and Byeong-Kweon Oh Seoul National University, Seoul National University Abstract : 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 Downloads: Full-text PDF