Shifted-prime divisors

Let ${\omega }^{\ast }(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of ${\omega }^{\ast }(n)$ are still somewhat a mystery. This paper addresses these higher moments and considers other related problems.