On the Number of Divisors of Mersenne Numbers

Denote f(n) := Sigma(1 <= k <= n) tau(2(k) - 1), where tau is the number of divisors function. Motivated by a question of Paul Erdos, we show that the sequence of ratios f(2n)/f(n) is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we give experimental evidence on both the conjecture f(2n)/f(n) ->infinity and our sufficient conditions for it to hold