  It has long been known that sequences such as the powers of $2$ and the
factorials satisfy Benford's Law; that is, leading digits in these sequences
occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In
this paper, we consider the leading digits of the Mersenne numbers
$M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime. In light of known
irregularities in the distribution of primes, one might expect that the leading
digit sequence of $\{M_n\}$ has \emph{worse} distribution properties than
"smooth" sequences with similar rates of growth, such as $\{2^{n\log n}\}$.
Surprisingly, the opposite seems to be the true; indeed, we present data, based
on the first billion terms of the sequence $\{M_n\}$, showing that leading
digits of Mersenne numbers behave in many respects \emph{more regularly} than
those in the above smooth sequences. We state several conjectures to this
effect, and we provide an heuristic explanation for the observed phenomena
based on classic models for the distribution of primes.
