Common divisors of the Euler function at related arguments

Let φ denote the Euler function. For a fixed integer k ≠ 0, we study positive integers n for which the largest prime factor of φ(n) also divides φ(n + k). We obtain an unconditional upper bound on the number of such integers n ≤ x, as well as unconditional lower bounds in each of the cases k > 0 and k <0. We also obtain some conditional lower bounds, for example, under the Prime K-tuplets Conjecture. Our lower bounds are based on explicit constructions.