As long as you store the hard boiled eggs in the refrigerator the hard boiled eggs will last up to 7 days.
As long as you kept the hard boiled eggs in the refrigerator for the week the hard boiled eggs will remain safe to eat a week later after you boiled them.
That goes for both peeled hard boiled eggs and unpeeled hard boiled eggs.
After a week they might be safe to eat but I would not chance it because they could have grown bacteria in them and if the hard boiled eggs have gone bad they would smell bad or have a bad taste to them.
The smell of a bad hard boiled egg would likely make your stomach turn and make you want to vomit.
If the hard boiled eggs taste okay or smell okay after a week then you can eat them and the worst that will happen is you get a stomach ache and food poisoning that will usually not be harmful and go away within a few days to a week.