Most common number of prime factors

I wrote some Python code to check what the most common number of prime factors is for all numbers between $1$ and $1,000,000$ (it took a while!). For example, $10$ has two prime factors, $2$ and $5$ , and $220$ has four: $2$ , $2$ , $5$ , and $11$ . Here's what I got:

(Please forgive the odd x-axis scale. I forgot to set this before I ran the program.)

At first I checked numbers between $1$ and $10,000$ , and then $1$ and $100,000$ , and when I did this, it appeared that it was most common for a number to have $2$ prime factors. But as I increased the upper bound, $3$ prime factors became the most frequent.

I'm not sure if this would change once again with an even larger upper bound, because my laptop took about 20 minutes to do up to $1,000,000$ . Does anyone have anything to add?

#NumberTheory

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Comments

You could ask @Zakir Husain, @David Stiff.

Yajat Shamji - 11 months, 2 weeks ago

@David Stiff - I have organised a discussion in which a code is share, we will execute it but by breaking into pieces (together) as it will take much longer time if executed by me alone. For more information click here, hope we will go beyond $10^9$ !

Zakir Husain - 11 months, 1 week ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$