The $n$-th Mersenne number is $M_n=2^n-1$. Let $A$ be the set of squarefree positive integers $a$ such that $M_n=a b^2$ for some positive integers $n$, $b$. My question is regarding the natural density of $A$, defined as $$ \delta_A=\lim_{X \rightarrow \infty} \frac{\# \{a \in A | a \le X\}}{X}. $$

Question: Show that $\delta_A=0$.

Edit: Writing $M_n=a_n b_n^2$ with $a_n$ squarefree, it is easy to show using the ABC conjecture that $b_n$ is negligible, and this implies that $\delta_A=0$. I'm looking for an unconditional proof.