On a result of Flammenkamp-Luca concerning noncototient sequence

Abstract

Let $\varphi(n)$ be the Euler totient function of $n$. A positive integer $m$ is called a noncototient if the equation $n-\varphi(n)=m$ has no solution in positive integers $n$. The sequence $(2^{k}p)_{k=1}^{\infty}$ which is noncototient for some prime $p$ will be called as Sierpiński's sequence. In this paper we prove some interesting properties of the Sierpiński sequence given in the Theorem 1, 2, 3.