I just thought about these two:
Logarithmic primes: Any natural number $n$ such that for any other natural number $a$ not equal to $n$, $\log_an$ is not an integer. That should be our logarithmic primes.
Root primes: Any natural number $n$ such that for any other natural number $a$ not equal to 1, $n^{\frac{1}{a}}$ is not an integer. That should be root primes.
I searched on the internet to see if the terms already existed but I didn't find them. So, do these terms exist? And why doesn't anyone talk about them? If they exist, they aren't popular. Are these sequences of numbers useless?