Looking at the various properties given we see that one property given is whether a number is happy. Happy Numbers makes a great investigation for students – they see the value in being systematic and recording their findings carefully as careful recording can save them much work as they develop the investigation. (The Happy Numbers starter above comes the Starter of the Day site. from Transum Software.)
A site some of my students like is Amazing Number Facts from Madras College.
See the old MAA NumberADay blog for many interesting number posts.