Let $N < 2^a$ be a positive integer chosen uniformly at random. Let $\tilde{N}$ be the result of removing from $N$ all its prime factors less than $2^b$. What is the probability that $\tilde{N}$ is composite and $\tilde{N} > 2^c$?

Context

I want to build the smallest-possible safe RSA modulus without a trusted party. A positive integer is a safe RSA modulus if, after removing all its prime factors less than 512 bits, it is composite and has size at least 2048 bits.

(That is we set $b=512$ and $c=2048$ in the above problem. The parameter $b=512$ protects against the ECM which has found primes of size up to 273 bits. The parameter $c=2048$ protects against the GNSF which factored numbers up to 768 bits.)

The strategy is to randomly sample several random numbers $N_1, ..., N_k$ and multiply them together. Each $N_i$ has some probability $p$ of being a safe RSA modulus so the product $N_1...N_k$ has probability $1 - (1-p)^k$ of being safe.

To choose $k$ appropriately I need a reasonably tight lower bound for $p$. (The above strategy of multiplying randomly chosen integers was pioneered by Thomas Sanders but he used a different—unnecessarily strict—definition of a safe RSA modulus.)