Find the largest positive integer $n<100$, such that there exists an arithmetic progression of positive integers $a_1,a_2,...,a_n$ with the following properties.
$1)$ All numbers $a_2,a_3,...,a_{n−1}$ are powers of positive integers, that is numbers of the form $j^k$, where $j \geq 1$ and $k \geq 2$ are integers.
$2)$ The numbers $a_1$ and $a_n$ are not powers of positive integers.
From a quick search, this question appears to still be open. Bounds on $n$ exist for some restricted sets of powers $k$, and it is known that the number of progressions of length $n$ for $n\geq 6$ with relatively coprime terms (if any) is finite.
See for instance:
"Arithmetic progressions consisting of unlike powers"
"Arithmetic progressions of squares, cubes, and $n$-th powers"