Let N be a positive integer. Is there an efficient (i.e. probabilistic polynomial time) algorithm which, on input a sufficiently large N, outputs the full factorization of some integer in the interval $[N - O(\log N), N]$?
Note that the running time of the algorithm is measured in $|N| = O(\log N)$.
Extra: The idea is to generate a factored integer as closed to N as possible. So, it is more desirable to have an efficient algorithm which, for large enough N, its output is a factored integer in the interval $[N - O(\log\log N), N]$.