In this paper, we give the order of magnitude for the summatory function of the number of divisors of the least common multiple of $p^i − 1$ for $i = 1, 2, . . . , k$ when $p le x$ is prime.
Luca, F., Pappalardi, F. (2021). On the number of divisors of the least common multiples of shifted prime powers. FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI, 64(2), 153-161.