The spanning method and the Lehmer totient problem

In this paper, we introduce and develop the notion of spanning of integers along functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to a class of problems that requires to determine if the equations of the form $tf(n)=n-k$ has a solution $n\in \mathbb{N}$ for a fixed $k\in \mathbb{N}$ and some $t\in \mathbb{N}$. In particular, we show that \begin{align} \# \{n\leq s~|~t\varphi(n)+1=n,~\mathbf{for~some}~t\in \mathbb{N}\}\geq \frac{s}{2\log s}\prod \limits_{p | \lfloor s\rfloor }(1-\frac{1}{p})^{-1}-\frac{3}{2}e^{\gamma}\nonumber \end{align} as $s\longrightarrow \infty$, where $\varphi$ is the Euler totient function and $\gamma=0.5772\cdots$ is the Euler-Macheroni constant.