KöMaL 2019 B5045

For which positive integers $n$ is there an appropriate order $a_1,a_2,\dots,a_n$ of the first $n$ positive integers such that the numbers $a_1+1,a_2+2,\dots,a_n+n$ are all perfect powers? (A number is called a perfect power if it can be represented in the form $a^b$, where $a, b\ge 2$ are integers.)