# Understanding a plot of composite numbers against the ordinal position of their prime factors

I was toying with primes factors of natural numbers and I have found a graph which caught my interest but one, which I am struggling to understand better.

Let us take the composite number $N$=391721.

• First, find its prime factors which are: 11, 149, 239.
• Next, note the ordinal number of each prime factors. Here for example:

1. 11 is the 5th prime
2. 149 is the 35th prime
3. 239 is the 52nd prime.

Then plot on the x-y plane: for each composite number $N$, Plot $N^{e/\pi}$ on the $x$-axis and the ordinal position of each of its prime factors on the $y$ axis. So in the example above, I get the three pairs:

1. x=391721^(E/PI) , y=5
2. x=391721^(E/PI) , y=35
3. x=391721^(E/PI) , y=52

This yields the following graph: So: x-axis is the natural number line ^ (E/PI) y-axis is the order of the prime factors

Question:

1. how do I interpret the graph based on what is known about Prime number distribution.
2. Is the steepest line observed in the graph the steepest possible line or is this the steepest line observed at this scale
3. Another way to frame "2" is that is there a known upper bound on (order-of-prime)/numeric-value-of-prime

Thanks.

• What's the E/PI for? Where do you get the 140 and 229? What do 5, 35 and 52 have to do with the graph? – Robert Israel Mar 9 '14 at 7:54
• Hi @RobertIsrael, I was trying to get the lines to be as linear as possible so I was trying to pick a ration and I tried E/PI and it seemed to get me the most linear graph. – user1172468 Mar 9 '14 at 16:37
• @RobertIsrael so the y axis the the order of the prime like 2 is 1, 3, is 2, ..., 11 is 5, 149 is 35, etc. -- I'll update the question with this clarificaiton – user1172468 Mar 9 '14 at 16:39
• So those y values should be 5, 35 and 52, not 11, 140 and 229? – Robert Israel Mar 9 '14 at 18:15
• @RobertIsrael -- yes to " 5, 35 and 52, not 11, 140 and 229?" -- I just realized my question text was wrong -- corrected -- thanks – user1172468 Mar 9 '14 at 18:30

So with $c = e/\pi$ you're plotting $[n^c, y]$ whenever $p_y$ divides $n$, where $p_y$ is the $y$'th prime.
The $k$'th curve from the top comes from the points where $n = k p$ where $p$ is prime (i.e. in the top curve $n$ is prime, in the second it's twice a prime, etc). Thus this is a plot of $[n^c, \pi(n/k)]$ where $\pi(x)$ is the number of primes $\le x$. Asymptotically, $\pi(x) \sim x/\ln(x)$, but over the range you're plotting the ratio of this to $x^c$ is not too far from constant.