TR-MO 2018 Problem 3

A sequence $a_1,a_2,\dots$ satisfy $$\sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10},$$ for every $n\in\mathbb{N}$ . Let $c$ be a positive integer. Prove that, for every positive integer $n$ , $$\frac{c^{a_n}-c^{a_{n-1}}}{n}$$ is an integer.