While doing recreational mathematics, I noticed a curious pattern related to the prime factorization of numbers. If you arrange this factorization for all the numbers up to a certain $N$, according to the table below,
One can add up all the exponents of each prime. By plotting the pairs (in the case of $N=100000$ in the image above) (2,99989), (3,49995), (5,24994), ... in a graph, it produces a curve that almost perfectly fits the equation $y=N/x$:
(Here is the graph in logarithmic scale for N=1000000 for primes up to 10000: https://i.stack.imgur.com/zPTpa.png)
In the image above, the red dots are the plotted points/pairs and the blue curve is the said equation.
In this sense, the above facts can be more specifically stated as the following problem:
For any number $n$, $$n=\prod_{i=1}^{\infty}p_i^{a_{n,i}}$$ Show that $$\sum_{i=1}^{N}a_{i,k}\sim N/p_k$$ Where $p_k$ is the kth prime.
Is this true, and are there useful/interesting implications?