A generalization of the Titchmarsh divisor problem

Let d⁽ᵏ⁾(n) be the k-free divisor function for integer k≥2. Let a be a nonzero integer. In this paper, we establish an asymptotic formula {equation*} ∑p≤ x d⁽ᵏ⁾(p-a) =b_k · x+O(x/log x) {equation*} related to the Titchmarsh divisor problem, where bₖ is a positive constant dependent on k and a. For the proof, we apply a result of Felix and show a general asymptotic formula for a class of arithmetic functions including the unitary divisor function, k-free divisor function and the proper Pillai's function.