I was trying to figure out how factoring out small primes first affected the run time of a more expensive factoring algorithm, whose run time is $O(f(n))$, where $f$ is some increasing function (though, in case it matters, $f(n)=e^{(1+o(1))\sqrt{ln(n)ln(ln(n))}}$). To this end, I wanted to find the expected value of $f(g_B(n))$ over some interval, where $g_B(n)=1$ if at most one of its prime factors is greater than $B$, and otherwise is $n/d$ where $d$ is the largest divisor of $n$ whose prime factors are all less than $B$.
I could do a much cruder analysis by simply finding the approximate number of $B$-smooth and prime numbers in the interval, but my intuition tells me that we end up removing a large enough factor often enough (without removing everything) that the more detailed analysis is worthwhile.
Has anybody seen an analysis of this sort? Is there a decent way to pull it off.