90

In this question I plotted the number of numbers with $n$ prime factors. It appears that the further out on the number line you go, the number of numbers with $3$ prime factors get ahead more and more.

The charts show the number of numbers with exactly $n$ prime factors, counted with multiplicity: enter image description here enter image description here (Please ignore the 'Divisors' in the chart legend, it should read 'Factors')

My question is: will the line for numbers with $3$ prime factors be overtaken by another line or do 'most numbers have $3$ prime factors'? It it is indeed that case that most numbers have $3$ prime factors, what is the explanation for this?

user21820
  • 53,491
  • 7
  • 84
  • 231
  • 4
    Fascinating graphs! It’s reasonably intuitive that numbers with a smaller number of factors would outpace those with more factors — because the number of primes is infinite and "evenly spaced" (cf. Prime Number Theorem)… but why it is, in order, 3-2-4-5-1, is a bit of a wonderful mystery. I’m curious what you find out. – Kieren MacMillan Feb 12 '20 at 14:59
  • 3
    The statements "$P(X \text{ has $3$ prime factors})>P(X \text{ has $k$ prime factors for some } k\ne3)$", and "$P(X \text{ has $3$ prime factors})>P(X \text{ has $k$ prime factors})$ for every $k\ne3$" (say, when $X\le N$ for some large $N$) are different. What the question title talks about is more or less the former, while the graph suggests the latter. – user1551 Feb 12 '20 at 15:09
  • 2
    @ SmallestUncomputableNumber: Maybe you can use a double-logarithmic plot, i.e. not plot $(n,\mathit{divisors})$ but instead $(\log n,\log(\mathit{divisors}))$. The current plot tricks you in thinking the growths were linear-ish. – emacs drives me nuts Feb 12 '20 at 15:09
  • @emacsdrivesmenuts Great idea, I will add them later – SmallestUncomputableNumber Feb 12 '20 at 15:26
  • @SmallestUncomputableNumber: For the current range, not much will change, but you have more room for bigger values. – emacs drives me nuts Feb 12 '20 at 15:32
  • If we plot the number of $10^3$ random integers $n$ up to $10^{40}$ having $k$ prime divisors (without multiplicity, i.e. $\omega(n)=k$), we get the [following plot](https://i.stack.imgur.com/nFFYv.png) (which I made on CoCalc using results from MAGMA) – the $x$ axis is $k$ and $y$ represents the number of $n ≤ 10^{40}$ (from a random sample of size $10^3$) with $\omega(n)=k$. We see that most of the numbers had 4 prime divisors ! We can observe a beautiful Gaussian! – Watson Feb 13 '20 at 15:01
  • 1
    Notice that we needed to search very far (up to $10^{40}$, but actually something like $10^{24}$ might be sufficient but would not be obvious on the plot), because of the $\log \log n$ term in Erdös–Kac theorem, which grows very slowly (also, here $\log$ denotes the natural logarithm). – Watson Feb 13 '20 at 15:01

2 Answers2

95

Yes, the line for numbers with $3$ prime factors will be overtaken by another line. As shown & explained in Prime Factors: Plotting the Prime Factor Frequencies, even up to $10$ million, the most frequent count is $3$, with the mean being close to it. However, it later says

For $n = 10^9$ the mean is close to $3$, and for $n = 10^{24}$ the mean is close to $4$.

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

OEIS A$001221$'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 $a(n): \sum_{k=1}^n a(k) \sim \sum_{k=1}^n \log \log k.$ - 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 $\omega(n)$ of distinct prime factors of a number $n$ is $\log(\log(n))$.

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

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

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 $10$ million.

PNT
  • 3,022
  • 6
  • 24
John Omielan
  • 39,427
  • 2
  • 26
  • 66
  • 5
    You can also just estimate the number of integers up to X with at most 2 prime factors, and show this is 0% of all integers. – Kimball Feb 13 '20 at 00:24
20

Just another plot to about $250\times10^9$, showing the relative amount of numbers below with x factors (with multiplicity) enter image description here

Somewhere between $151,100,000,000$ and $151,200,000,000$ 4 overtakes r3.

PNT
  • 3,022
  • 6
  • 24
Michael Stocker
  • 1,154
  • 6
  • 12
  • See also my comment [above](https://math.stackexchange.com/questions/3544034/do-most-numbers-have-exactly-3-prime-factors/3547453#comment7290820_3544034). – Watson Feb 16 '20 at 09:49