On the least common multiple of shifted powers

Let $a\ge 2$ be an integer. We prove that for every periodic sequence $(s_n{)}_{n\ge 1}$ in $\{-1,+1\}$ there exists an effectively computable rational number $C_{\mathbf{s}}>0$ such that \begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim C_$\mathbf{s}$ \cdot \frac{\log a}{\pi^2} \cdot n^2 , \end{equation*} as $n\to +\mathrm{\infty }$, where $\mathrm{lcm}$ denotes the least common multiple. Furthermore, we show that if $(s_n{)}_{n\ge 1}$ is a sequence of independent and uniformly distributed random variables in $\{-1,+1\}$ then \begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim 6 \operatorname{Li}_2!\big(\tfrac1{2}\big) \cdot \frac{\log a}{\pi^2} \cdot n^2 , \end{equation*} with probability $1-o(1)$, as $n\to +\mathrm{\infty }$, where ${\mathrm{Li}}_2$ is the dilogarithm function.