The reciprocal sum of divisors of Mersenne numbers

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 ${\max }_{n\le x}{\sum }_{p\mid 2^n-1}1\mathrm{/}p$ to within $o(1)$ and ${\max }_{n\le x}{\sum }_{d\mid 2^n-1}1\mathrm{/}d$ to within a factor of $1+o(1)$, as $x\to \mathrm{\infty }$. This refines, conditionally, earlier estimates of Erd\H{o}s and Erd\H{o}s-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $\sum 1/d$towithinafactorof$1+o(1)$where$d$runsoverallnumbersdividing$2^n-1$forsome$n\le x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both ${\sum }_{p\mid 2^n-1}1\mathrm{/}p$ and ${\sum }_{d\mid 2^n-1}1\mathrm{/}d$ admit continuous distribution functions in the sense of probabilistic number theory.