Happy (and unhappy) numbers

While watching the Dr. Who episode from season 3 titled "42" last night, David Tennant rattled off:

"Any number that reduces to one when you take the sum of the square of its digits, reduce it and continue iterating until it yields 1, is a happy number. Any number that doesn't, isn't. A happy prime is both happy and prime"

As I've never taken a recreational mathematics course, in my mind's eye all I saw was this:

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