So if we try to categorize numbers based on the number of their prime factors we would have something as following where $L_n$ is the list of numbers with $n$ prime factors. $$ L_1 : 2, 3, 5, 7, 11, ... $$ $$ L_2 :2^2, 2 \times 3, 3^2, 2\times5, ... $$
$$ ... $$
$$ L_n : 2^n, ...,3^n, ..., 5^n, ...p^n $$
Here on $L_n$ the dots show the products of primes where the sum of their powers is $n$. For example between $2^3$ and $3^3$ there is $2 \times\ 2 \times 3$ , $2 \times\ 3 \times 3$ , etc..
My question now is whether there is a formula that can give the amount of members of $L_n$ between $p_i^n$ and $p_{i+1}^n,$ in terms of $i$ and $n$? Or since it's obvious that there must at least be $n-1$ numbers between them, what's the upper bound?
In any interval, we can correctly estimate how many prime-numbers in this interval, and alse how many integers with $k$ factors.
I don't know any theory on this, so I did some tests.
Between $500 000 000$ and $510 000 000$ :
Numbers with 1 factor are 4.74% ; 2 factors : 15.9% 3:22.88% 4:21.3% 5:15.2% 6:9.25% 7:5.14% 8:2.68% 9:1.37% 10:0.66% 11:0.33% 12:0.16%
And divide by 2 for any following value.
Between $730 000 000$ and $740 000 000$
Numbers with 1 factor are 4.66% ; 2 factors : 15.69% 3:22.78% 4:21.3% 5:15.34% 6:9.34% 7:5.21% 8:2.73% 9:1.39% And divide by 2 for any following value.
Proportion of numbers with 5 factors or more increase very slowly.
So, when you know the length of the interval between $p_i^n$ and $p_{i+1}^n$, you can estimate how many integers with $k$ factors in this interval. It doesn't work if $p_i$ is very small (2, 3, 5 ... 53 ...)