Asymptotic law of distribution of primes of special form

Let N₀ be the set of natural numbers whose binary expansions have an even number of 1’s, and let N₁ = N\N₀. In this paper, we obtain asymptotic formulas for the number of primes p not exceeding X and such that p ∈ N_i, p + 1 ∈ N_j, where i and j take values 0 and 1 independently of each other.