Overpseudoprimes, and mersenne and Fermat numbers as primover numbers

We introduce a new class of pseudoprimes, that we call "overpseudoprimes to base b", which is a subclass of the strong pseudoprimes to base b. Letting |b|_n denote the multiplicative order of b modulo n, we show that a composite number n is an overpseu- doprime if and only if |b|_d is invariant for all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2^p - 1, where p is prime, are overpseudoprimes to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well-known numbers are "primover to base b"; i.e., they are primes or overpseudoprimes to base b.