As we all know, a natural number $n$ is prime if and only if there do not exist natural numbers $x, y$ exclusively between $1$ and $n$ such that $xy = n$.
Is there any generally recognized analogy for primes in powers? For example, a natural number $n$ in a set $S$ that is not the $x$th power of $y$ for two natural numbers $x$ and $y$ exclusively between $1$ and $n$? I recognized that the definition is less elegant than that for multiplication, since powers aren't commutative, but I was curious if this set or any similar set has been explored in mathematics.
For clarity, in my example definition, $1, 2, 3, 5, 6, 7 \in S$, but $4 = 2^2, 8 = 2^3, 9 = 3^2 \not\in S$.