Quote:
Originally Posted by R.D. Silverman
Wrong approach. What needs to be computed is the conditional probability that a large integer N has k prime factors given that it has no factors less than (say) N^1/a, for given a.

That's a good approach for the scenarios where you can calculate that distribution  you can start directly with the residual composite. The only way I know to calculate that distribution is to use the inclusionexclusion and normalization approach described in your paper with Wagstaff. That works for cases where the total number of factors is a handful. But some cases of interest will have "a" of hundreds or thousands.
For these scenarios, I think you would be better off to start with the entire original number and the distibutions I described, then use Bayes to account for the known factoring efforts.
But it doesn't really matter what I think or you think  this is a question of comparing heuristics that can studied empirically. Perhaps we can get the OP to propose some "interesting" cases.