A k-bit number is a positive natural number with k bits when expressed in binary, i.e. a number less than 2^k and at least 2^(k-1)

This question asks: What fraction of composite 6144-bit numbers with no prime factors with 30 bits or less have at least two prime factors 1024 bits or more?

This question is relevant to an upcoming blog post of mine. It resolves when I manage to run code that statistically guarantees (to confidence level 10^-15, let us say), which bucket the answer falls in.

Writing such code should be straightforward using a pre-factored numbers procedure. I have a draft of code that samples from the Bernoulli distribution with this parameter, and it looks like it would return a sample every 100 days or so running on my laptop. I am fairly confident that with algorithmic improvements or enough compute it could be run fast enough to get an answer in a reasonable amount of time.