An ErdŐs–Kac theorem for integers with dense divisors

Abstract

We show that for large integers n, whose ratios of consecutive divisors are bound 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 $C=1/(1-{\rm e}^{-\gamma})\approx 2.280$ and V ≈ 0.414. This result is then generalized in two different directions.