Let $a_1,a_2,a_3,...$ be an infinite sequence of positive integers. Suppose that a sequence $a_1,a_2,\ldots$ of positive integers satisfies $a_1=1$ and \[a_{n}=\sum_{n\neq d|n}a_d\]for every integer $n>1$. Prove that the exist infinitely many integers $k$ such that $a_k=k$.