Probability that two integers have the very same amount of prime factors

Question

I am looking forward to finding this probability when these 2 integers are uniformely and independantly chosen between 1 and n given.

Answer 1

Tough question.

We know only (Erdős-Kac theorem) that if $\omega(n)$ denotes the number of distinct prime factors of $n$ and if we choose a random $n$ between $1$ and $N$ then

$$\frac{\omega(n)-\log\log N}{\sqrt{\log\log N}}$$

is a standard normal random variable if $N$ is large. (Better said: it tends to that if $N \to\infty$.)

You can use this result to estimate the probabilty in question in case of large $N$s.