Is it permissible for a student to be absent for four weeks, given that the first week is not usually counted as an absence, and knowing that the university system allows absence for only three weeks?

So long as the prevailing custom in the university environment is that the first week of study does not involve attendance taking or recording of absences, there is no sin upon one who is absent from it. This is because it is common for professors not to record absences without explicit permission granted to them by the administration.