Let a prime power \(p^n\) divide a Mersenne number \(M_q=2^q-1\). It is shown here that \(p\) is a Wieferich prime of order \(n\) if and only if \(p^{n+1}\) divides \(M_q\) or equivalently, if and only if the number 2 has multiplicative order \(q\) modulo \(p^{n+1}\). Recall that a Wieferich prime is characterized by \(2^{p-1}\equiv 1\pmod {p^2}\). In this generalization, a Wieferich prime of order \(n\) is a Wieferich prime for which \(2^{p^n-p^{n-1}}\equiv 1 \pmod {p^{2n}}\).