An Erdős-Kac theorem for integers with dense divisors

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 $C=1\mathrm{/}(1-e^{-\gamma })\approx 2.280$ and $V\approx 0.414$. This result is then generalized in two different directions.