Reversible primes

For an n-bit positive integer a written in binary as a = ∑ⱼ₌₀ⁿ⁻¹ εⱼ(a) \,2ʲ where, εⱼ(a) ∈ \0,1\, j∈\0, …, n-1\, εₙ₋₁(a)=1, let us define a = ∑ⱼ₌₀ⁿ⁻¹ εⱼ(a)\,2ⁿ⁻¹⁻ʲ, the digital reversal of a. Also let Bₙ = \2ⁿ⁻¹≤ a<2ⁿ:~a odd\. With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of p ∈ Bₙ such that p and p are prime. We also prove that for sufficiently large n, | ∈ Bₙ:~ max \Ω(a), Ω(a)\≤ 8 \| ≥ c\, 2ⁿ/n², where $Ω(n)$ 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 a are both squarefree. Our method leads us to provide various estimates for the exponential sum ∑a ∈ Bₙ exp(2πi (αa + ϑ a)) (α,ϑ ).