Estimates for Wieferich numbers

We define Wieferich numbers to be those odd integers w >= 3 that satisfy the congruence 2(phi(w)) equivalent to 1 (mod w(2)). It is clear that the distribution of Wieferich numbers is closely related to the distribution of Wieferich primes, and we give some quantitative forms of this statement. We establish several unconditional asymptotic results about Wieferich numbers; analogous results for the set of Wieferich primes remain out of reach. Finally, we consider several modifications of the above definition and demonstrate that our methods apply to such sets of integers as well.