On a problem of S. Ramanujan (Q1763519)

Denote by \(d(n)\) the number of divisors of a positive integer \(n\) and by \(\omega(n)\) the number of distinct prime divisors of \(n\). The problem of investigating the iterated function \(d(d(n))\) goes back to S. Ramanujan in 1915, and, in [Bull., Cl. Sci. Math. Nat., Sci. Math. 17, 13--22 (1989; Zbl 0695.10040)], \textit{P. Erdös} and \textit{A. Ivić} established an upper bound for it that is valid for all sufficiently large \(n\). The present author improves this upper bound by first deriving an upper bound for \(\omega(d(n))\). He also obtains a lower bound for \(\omega(d(n))\) that holds for infinitely many \(n,\) this leads to a lower bound for \(d(d(n))\) holding for infinitely many \(n\) that improves Ramanujan's own lower bound. The proofs are combinatorial and elementary in nature.