The number of prime factors of integers with dense divisors

We show that for integers n, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is Clog⁡log⁡n, where C=(1−e−γ)−1=2.280... and γ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.