We claim that there exists such $a$. Let $a=2015^{x_1}2016^{x_2}\cdots2558^{x_{544}}$. By CRT, there exist positive integers $(x_i)^{544}_{i=1}$ such that $$x_i\equiv\begin{cases} 0\pmod{p_j} &\text{if } j\neq i. \\ -1\pmod{p_j} &\text{if } j=i. \end{cases}$$for every $1\leq i\leq 544$ and prime number $p_1<p_2<\cdots<p_{544}$. Thus $(2014+i)(a)$ is the $p_i^\text{th}$ power of some positive integer for every $1\leq i\leq 544$.