Problem

Source: Peru Cono Sur TST 2020 P6

Tags: number theory



Let $a_1, a_2, a_3, \ldots$ a sequence of positive integers that satisfy the following conditions: $$a_1=1, a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor}, \forall n\ge 1$$Prove that for every positive integer $k$ there exists a term $a_i$ that is divisible by $k$