How to prove that an integer is hard to factorize when sampled from a known distribution
On trail to follow, from Handbook of Applied Cryptography fact 3.7:

Let $n$ be chosen uniformly at random form the interval $[1, x]$.


if $1/2 \leq \alpha \leq 1$, then the probability that the largest prime factor of $n$ is $\leq x^{\alpha}$ is approximately $1+ ln(\alpha)$. Thus, for example, the probability than $n$ has a prime factor $&gt; \sqrt(x)$ is $ln(2) \approx 0.69$
The probability that the second-largest prime factor of $n$ is $\leq x^{0.2117}$ is about $1/2$. 
The expected total number of prime factors of $n$ is $ln ln x + \mathbb{O}(1)$. (If $n = \prod p_i^{e_i}$, the total number of prime factors of n is $\sum e_i$.)