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Integrality of a certain quantity

Source: Turkey National Mathematical Olympiad 2018

December 2, 2018
Sequencesalgebranumber theoryTurkey

Problem Statement

A sequence a1,a2,…a_1,a_2,\dots satisfy ∑i=1na⌊ni⌋=n10, \sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10}, for every n∈Nn\in\mathbb{N}. Let cc be a positive integer. Prove that, for every positive integer nn, can−can−1n \frac{c^{a_n}-c^{a_{n-1}}}{n} is an integer.
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