$\omega(n)$ is the number of distinct prime factors of $n$ and $\omega'(n)$ is the number of distinct prime factors of $n$ with multiplicity. For example if $p,q$ are prime numbers then $\omega(p^2q)=2$ and $\omega'(p^2q)=3$.
What is the distribution of $\omega'(n)$ for $n$ between $2^m$ and $2^{m+1}$ where $m$ is a sufficiently large integer?
Do we have with probability at least $\frac12$ at least $\log\log n$ indistinct factors?