How to prove that an integer is hard to factorize when sampled from a known distribution Suppose that an $n$-bit integer $c$ has been randomly drawn from a distribution $\chi$, whose description is known. Is there a general method to check if this particular sampling helps factorize $n$? Are there any results in this direction (i.e. results like "don't choose an RSA modulus whose factors were sampled with this or that 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 $\sum e_i$.)