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In this paper we introduce and develop the notion of spanning of integersalong functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this methodto a class of problems requiring to determine if the equations of the form$tf(n)=n-k$ has a solution $n\in \mathbb{N}$ for a fixed $k\in \mathbb{N}$ andsome $t\in \mathbb{N}$. In particular, we show that \begin{align} \# \{n\leqs~|~t\varphi(n)+1=n,~t,n \in \mathbb{N}\}\geq \frac{s}{2\log s}\prod \limits_{p| s}(1-\frac{1}{p})^{-1}+O(1)\nonumber \end{align}where $\varphi$ is the eulertotient function.

The spanning method and the Lehmer totient problem