The number of prime factors of integers with dense divisors (Q2671993)

A positive integer \(n\) is \(t\)-dense if the ratios of consecutive divisors of \(n\) do not exceed \(t\). Let \(\mathcal{D}(x,t)\) be the set of \(t\)-dense integers \(n\leq x\), and \(\omega(n)\) (resp. \(\Omega(n)\)) be the number of prime factors of \(n\), counted without (resp. with) multiplicity. In the paper under review, the author proves that uniformly for \(x\geq t\geq 2\), \[ \frac{1}{|\mathcal{D}(x,t)|}\sum_{n\in\mathcal{D}(x,t)}\omega(n)=C\log\log x-(C-1)\log\log t+O(1), \] where \(C=1/(1-e^{-\gamma})\). The same result also holds with \(\Omega\) in place of \(\omega\). The author generalizes this result to a set covering the cases of the set of \(t\)-dense integers and the set of practical numbers.