Generalized Cullen numbers with the Lehmer property (Q2872187)

A composite integer \(n\) is a Lehmer number if \(\varphi(n)\) divides \(n-1\), where \(\varphi\) is Euler's totient function. No Lehmer number is known. Clearly, \(\varphi(n)\) divides \(n-1\) for each prime number \(n\).NEWLINENEWLINEAn integer of the form \(C_n=n2^n+1\) is a Cullen number. Recently, \textit{J. M. Grau Ribas} and \textit{F. Luca} [Proc. Am. Math. Soc. 140, No. 1, 129--134 (2012; Zbl 1272.11003); Corrigendum ibid. 141, No. 8, 2941--2943 (2013; Zbl 1272.11004)] proved that every Cullen number is not a Lehmer number. Motivated by their proof, the present authors prove that there does not exist a Lehmer number of the form \(D_{p, n}:=np^n+1\), where \(p\) is prime. The authors also show that a positive integer of the form \(\alpha 2^{\beta}+1\), where \(\alpha\leq\beta\) and \(\alpha\) is odd, is not a Lehmer number.