On the number of divisors of Mersenne numbers

Denote $f(n):={\sum }_{1\le k\le n}\tau (2^k-1)$, where $\tau$ is the number of divisors function. Motivated by a question of Paul Erdős, we show that the sequence of ratios $f(2n)\mathrm{/}f(n)$ is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we test numerically both the conjecture $f(2n)\mathrm{/}f(n)\to \mathrm{\infty }$ and our sufficient conditions for it to hold.