  We count ]B, C]-grained, k-factor integers which are simultaneously B-rough
and C-smooth and have a fixed number k of prime factors. Our aim is to exploit
explicit versions of the prime number theorem as much as possible to get good
explicit bounds for the count of such integers. This analysis was inspired by
certain inner procedures in the general number field sieve. The result should
at least provide some insight in what happens there.
  We estimate the given count in terms of some recursively defined functions.
Since they are still difficult to handle, only another approximation step
reveals their orders.
  Finally, we use the obtained bounds to perform numerical experiments that
show how good the desired count can be approximated for the parameters of the
general number field sieve in the mentioned inspiring application.
