A note on the normal largest gap between prime factors

Let $\{p_j(n){\}}_{j=1}^{\omega (n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi (n)$ tending to infinity with $n$, we have $$f(n):=\underset{1⩽j<\omega (n)}{\max }\log \left(\frac{\log p_{j+1}(n)}{\log p_j(n)}\right)={\log }_{3}n+O(\xi (n))$$ for almost all integers $n$.