In my studies of how primes jump (search this forum for a link), a question has been raised which may have been studied. Can anyone jump-start my literature search with references regarding the following problem?

Let $\omega$ denote the number of distinct prime factors of a number $n$, and consider the minimal value $w$ of this number as $n$ ranges over the interval (N,N+D), where D depends on N and might be an iterated logarithm of N or a fractional power of N.

Problem: for all N and a given small function D of N, what provable upper bounds do we have on $w$?

Of course, for infinitely many N and D a small constant, we have $w=1$. I look for results that apply for all N, but am willing to accept $w=2$ or some other small constant, or even $w=$ log log log N. Even a careful analysis of multiples of $\omega$-small numbers would be welcome.

Gerhard "Trying To Jump To Conclusions" Paseman, 2016.10.25.