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

Question

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.

Answer 1

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.