Divisors of Mersenne numbers under doubling

This paper studies a question of Erdős about how the total number of divisors of Mersenne numbers grows when the range is doubled. We prove that the doubling ratio is always at least 3, sharp, and nearly 4 whenever the relevant Fermat numbers are composite, as all known ones are. We also reduce the open question of whether the ratio tends to infinity to a statement about indices with few divisors. This leads to a new conjecture on the prime factors of Mersenne numbers next to near-Wagstaff numbers, which we study through implications, barrier results and a proven polynomial analogue.