An Erd\H{o}s-Kac theorem for integers with dense divisors

Abstract

We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$, where $C=1/(1-e^{-\gamma})\approx 2.280$ and $V\approx 0.414$. This result is then generalized in two different directions.