An ErdŐs–Kac theorem for integers with dense divisors.

The article investigates the statistical distribution of the number of prime factors, denoted by \(\nu(n)\), of large integers \(n\) whose consecutive divisors satisfy strong multiplicative constraints, specifically those in the set \(\mathcal{D}(x,t)\) where the ratio of consecutive divisors is bounded by a parameter \(t\). It establishes an Erdős–Kac type theorem showing that \(\nu(n)\) follows an approximate normal (Gaussian) distribution with mean and variance proportional to \(\log_2 n\), but with a mean significantly larger than the variance, indicating a deviation from classical Poisson behavior. This result is generalized to a broader family of integer sequences \(\mathcal{B}_\vartheta\), defined via an arithmetic function \(\vartheta\), which includes practical numbers as a special case. The proofs rely on advanced analytic techniques involving functional equations, Laplace transforms, and delay differential equations related to Buchstab-type functions, culminating in precise asymptotic formulas for sums of \(z^{\nu(n)}\) over these constrained sets.