  In this paper we determined explicitly the multiplicative inverses of the
Dobbertin and Welch APN exponents in Z_{2^n-1}, and we described the binary
weights of the inverses of the Gold and Kasami exponents. We studied the
function \de(n), which for a fixed positive integer d maps integers n\geq 1 to
the least positive residue of the inverse of d modulo 2^n-1, if it exists. In
particular, we showed that the function \de is completely determined by its
values for 1 \leq n \leq \ordb, where \ordb is the order of 2 modulo the
largest odd divisor of d.
