# Do most numbers have exactly 33 prime factors?

In this question I plotted the number of numbers with $$nn$$ prime factors. It appears that the further out on the number line you go, the number of numbers with $$33$$ prime factors get ahead more and more.

The charts show the number of numbers with exactly $$nn$$ prime factors, counted with multiplicity:  (Please ignore the ‘Divisors’ in the chart legend, it should read ‘Factors’)

My question is: will the line for numbers with $$33$$ prime factors be overtaken by another line or do ‘most numbers have $$33$$ prime factors’? It it is indeed that case that most numbers have $$33$$ prime factors, what is the explanation for this?

Yes, the line for numbers with $$33$$ prime factors will be overtaken by another line. As shown & explained in Prime Factors: Plotting the Prime Factor Frequencies, even up to $$1010$$ million, the most frequent count is $$33$$, with the mean being close to it. However, it later says

For $$n = 10^9n = 10^9$$ the mean is close to $$33$$, and for $$n = 10^{24}n = 10^{24}$$ the mean is close to $$44$$.

The most common # of prime factors increases, but only very slowly, and with the mean having “no upper limit”.

OEIS A$$001221001221$$‘s closely related (i.e., where multiplicities are not counted) Number of distinct primes dividing n (also called omega(n)) says

The average order of $$a(n): \sum_{k=1}^n a(k) \sim \sum_{k=1}^n \log \log k.a(n): \sum_{k=1}^n a(k) \sim \sum_{k=1}^n \log \log k.$$Daniel Forgues, Aug 13-16 2015

Since this involves the log of a log, it helps explain why the average order increases only very slowly.

In addition, the Hardy–Ramanujan theorem says

… the normal order of the number $$\omega(n)\omega(n)$$ of distinct prime factors of a number $$nn$$ is $$\log(\log(n))\log(\log(n))$$.

Roughly speaking, this means that most numbers have about this number of distinct prime factors.

Also, regarding the statistical distribution, you have the Erdős–Kac theorem which states

… if $$ω(n)ω(n)$$ is the number of distinct prime factors of $$nn$$ (sequence A001221 in the OEIS, then, loosely speaking, the probability distribution of

$$\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}$$

is the standard normal distribution.

To see graphs related to this distribution, the first linked page of Prime Factors: Plotting the Prime Factor Frequencies has one which shows the values up to $$1010$$ million.