Calendar.app on both iOS and OS X provides functionality to add week numbers. However, I'd like the week numbers to match a custom pattern (numbered by semester rather than by year). Is there a way I can do that?
I do not know of any way of customising the existing function, so I suggest that you make an iCloud Calendar and make a repeated calendar entry for each semester. Example: Make new calendar named Semesters. Make an entry called 201516-1 and set the first Monday of the semester and repeat until the last. Continue making entries 201516-2, 201516-3, 201516-4 until you are done.