  In this work we present a natural generalization of strong pseudoprime to
base $b$, which we have called $q$-pseudoprime to base $b$. It allows us to
present another way to define a Midy's number to base $b$ (overpseudoprime to
base $b$). Besides, we count the bases $b$ such that $N$ is a $q$-probable
prime base $b$ and those ones such that $N$ is a Midy's number to base $b$.
Furthemore, we prove that there is not a concept analogous to Carmichael
numbers to $q$-probable prime to base $b$ as with the concept of strong
pseudoprimes to base $b$.
