  We investigate various questions concerning the reciprocal sum of divisors,
or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the
Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we
determine $\max_{n\le x} \sum_{p \mid 2^n-1} 1/p$ to within $o(1)$ and
$\max_{n\le x} \sum_{d\mid 2^n-1}1/d$ to within a factor of $1+o(1)$, as
$x\to\infty$. This refines, conditionally, earlier estimates of Erd\H{o}s and
Erd\H{o}s-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $\sum
1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing
$2^n-1$ for some $n\le x$. This conditionally confirms a conjecture of
Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show
that both $\sum_{p\mid 2^n-1} 1/p$ and $\sum_{d\mid 2^n-1}1/d$ admit continuous
distribution functions in the sense of probabilistic number theory.
