Proof of number of logins in last N days without saving login times
I'm not sure cryptography is the way to go. A day level counter could suffice, possibly aggregated later to coarser resolution and deleting any old information you no longer need.
If you want accurate counting for last N days with a certain resolution of last N days the information on what you did N days ago must be kept. It can be inferred by querying count and querying it again after a short while. But you could perhaps do things more fuzzy, maybe store some users data together and inactivate a user group if together they didn't meet the threshold and replace which users are grouped together periodically.