Prove that there are no positive integers $a_1, a_2, \ldots , a_{2021}$ (not necessarily distinct) such that for $k = 1, 2, 3, \ldots , 2021$ the number of elements in the set $$A_k = \{ j \in \mathbb{N} : 1 \le j \le 2021 \text{ and } a_j|k \}$$be exactly $a_k$.