Let $p$ be a prime. Is there a classification of the $k \geq 1$ such that $\gcd(k,p^k-1)=1$? If not, can we at least produce an explicit infinite subset? What is known about these $k$?

For the lack of a better name, let me call $k$ good for $p$ if $\gcd(k,p^k-1)=1$. For $p=2$ many numbers are good. This seems to be related but not identical to the OEIS sequence A049093. For $p > 2$ all good numbers are odd. For $p=3$ most odd numbers are good; for $k \leq 1000$ the only $24$ exceptions are 39, 55, 117, 165, 195, 253, 273, 275, 351, 385, 429, 495, 507, 585, 605, 663, 715, 741, 759, 819, 825, 897, 935, 975. There is no OEIS sequence for these. For $p=5$ it is similar, but for $p > 5$ the story becomes different.

Background. I can prove a certain theorem in abstract algebra for all pairs $(p,k)$ with $\gcd(k,p^k-1)=1$. Now, I wonder how many cases I have already covered. I am not a number theorist, so I apologize if this question turns out to be stupid.