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.

The $n$-th Mersenne number is $M_n=2^n-1$. Write $M_n=a_n b_n^2$ where $a_n$ is positive and squarefree.

Question 1: What lower bound can be proved for $a_n$?

Let $A$ be the set of all possible $a_n$. The natural density of $A$ is defined as $$ \delta_A=\lim_{X \rightarrow \infty} \frac{\# \{a \in A | a \le X\}}{X}. $$

Question 2: What can be proved about $\delta_A$? Is it possible to show that $\delta_A=0$?

Note: I am interested in unconditional answers to the above questions. It is easy to give answers conditional on the ABC conjecture. Indeed, the ABC conjecture shows that for any $\epsilon>0$ there is some $K_\epsilon>0$ such that $$ a_n \ge K_\epsilon \cdot 2^{n(1-\epsilon)}. $$ Thus $\# \{ a \in A | a \le X\}=O(\log(X))$, which gives $\delta_A=0$.

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 $\delta_A=0$.

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.