$\omega(n)$ 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?
I just discovered Matlab's factor()-function, and I randomly typed in 20081294819, and to my surprise it only had two factors (5099 and 3938281)! I had expected many more factors for such a big number and wondered how normal this was. The following is what I've done in my investigation so far.  In the first figure below, I have plotted the probability that some number below $10^k$ has $m$ indistinct prime factors.  The second figure shows the evolution of the probability of finding a number with $m$ indistinct prime factors as the numbers one is looking at grows, i.e. it is just another way to visualize the same data. I have left out all points of zero probability.  If one looks at the first part of the figure, it seems that the peak of the curve for some $k$ shifts toward higher $m$ as $k$ grows.   My first question is this: Does this behavior (i.e. the peak shifting toward higher $m$ as $k$ grows) continue indefinitely, or will the peak stabilize at some point? The latter seems unlikely but I can't be sure.  My second question is this: Approximately how fast does the peak-value $m_{peak}$ move as $k$ grows?  I briefly looked at the , but this involves distinct primes and I don't really know much number theory, so any help would be greatly appreciated!
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