Reversible primes

For an $n$-bit positive integer $a$ written in binary as $$a=\sum _{j=0}^{n-1}{\epsilon }_j(a)2^j$$ where, ${\epsilon }_j(a)\in \{0,1\}$, $j\in \{0,\dots ,n-1\}$, ${\epsilon }_{n-1}(a)=1$, let us define $$\overleftarrow{a}=\sum _{j=0}^{n-1}{\epsilon }_j(a)2^{n-1-j},$$ the digital reversal of $a$. Also let ${\mathcal{B}}_n=\{2^{n-1}\le a<2^n:a\text{ odd}\}\mathrm{.}$ With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of $p\in {\mathcal{B}}_n$ such that $p$ and $\overleftarrow{p}$ are prime. We also prove that for sufficiently large $n$, $$\mid \{a\in {\mathcal{B}}_n:\max \{\mathrm{\Omega }(a),\mathrm{\Omega }(\overleftarrow{a})\}\le 8\}\mid \ge c\frac{2^n}{n^2},$$ where $\mathrm{\Omega }(n)$ denotes the number of prime factors counted with multiplicity of $n$ and $c>0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of $n$-bit integers $a$ such that $a$ and $\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum $$\sum _{a\in {\mathcal{B}}_n}\exp (2\pi i(\alpha a+\vartheta \overleftarrow{a}))(\alpha ,\vartheta \in \mathbb{R})\mathrm{.}$$