# Is there a pattern for the distribution of prime factor count for numbers under n?

In the below picture I have charted the distribution of numbers below n by factor count. The bottom line is for all numbers under 100,000 then 200,000 ... all the way to 1,000,000.

They seem to tend to a specific set of numbers. Right now I am limited with my data set and was hoping someone could enlighten me as to if this goes on to infinity or does it tend to a specific number set?

The numbers in the chart below were calculated by:

Primes under 100,000 - 9,592 - I then split up how many had two factors and three and so on and divided those numbers by 9,592 and that produced the bottom line in the chart below.

The first number is always one because I divide the number of primes under n by that same number. Then the line rises because there are more composite numbers with two factors than there are prime numbers. It rises again with three and then starts to fall as the count of composites with 4 factors starts to trend in the other direction.

The reason I am doing this is for recreation prime research and with the thought that if we can see how the distribution of prime factors works with large numbers maybe this could develop a reasonable prime counting calculation.

• I'm having difficulty deciphering the graph. It seems all the curves start at $(1,1)$, and the horizontal axis presumably corresponds to some whole number of prime factors (according to multiplicity, or counting the number of distinct primes?). But the vertical axis has me baffled. – hardmath Mar 16 '15 at 1:08
• I edited it, let me know if that helps. I had to do the math this way so that I was comparing apples to apples, otherwise there would have been huge gaps that would have been difficult to read any discernible trend through. – Joe Mar 16 '15 at 1:23
• I suspected that the vertical axis was essentially normalizing to the number of primes in a range. This appears to give some convergence of the curves, though we cannot strictly say this is a "probability distribution". There are some related results in Knuth's AOCP vol. II Seminumerical Algorithms. I'll find some specific citations for you. – hardmath Mar 16 '15 at 4:49
• A variant of this question (asked by a different Joe) was answered by Carlo Beenakker on MathOverflow here. – Joseph O'Rourke Oct 5 '15 at 14:59