On the iteration of the number of divisors without large prime factors (Q1912302)

Let \(d(n)\) denote the number of divisors of \(n\) and \(d(d(n))\) be the iterated divisor function. Among others \textit{E. Heppner} [J. Reine Angew. Math. 265, 176-182 (1974; Zbl 0273.10037)] proved an asymptotic representation for the sum \(\sum_{n\leq x} d(d(n))\). The author considers the same sum, but under the restriction \(p(n)\leq y\leq x\), where \(p(n)\) denotes the greatest prime divisor of \(n\). Further, asymptotic results are given for the sum \[ \sum_{n\leq x} {d(d(n))\over p(n)} \] and similar sums.