As usual any related references are appreciated.

Let $p \lt q$ be distinct primes, and for all such pairs, let $m=pq$ and let $\cal{C}$ be the collection $(m-p,m)$ of open intervals. Does (the union of) this collection cover all but finitely many natural numbers?

Suppose we extend the collection to $\cal{D}$ by including $(m-p,m)$ where $m=p^a q^b$, so $m$ ranges over all natural numbers with $\omega(m)=2$ and $p$ is the smallest prime factor of $m$. Does $\cal{D}$ cover all but finitely many natural numbers?

Pick a natural number $t\gt 2$ and consider analogous collections of square free (or not) $m$ with $\omega(m)$ different from 1 and equal to (or at most ) $t$, again with $p$ prime and least dividing $m$, and again just using open intervals of the form $(m-p,m)$. Is there a $t$ such that one of these variations on $\cal{C}$ covers all but finitely many natural numbers?

Gerhard "Yes, It Concerns Jumping Primes" Paseman, 2016.11.02.