A number is called perfect if the sum of the proper divisors is equal to the number. 8 is not a perfect number because the proper divisors of 8 are 1, 2, and 4, and 1 + 2 + 4 = 7, which is not 8.

Can you find a perfect number? Can you find all of the perfect numbers less than 1000? Could you write a program to find all of the perfect numbers smaller than a million? Is there a largest possible perfect number?

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You might consider investigating the sum of divisors more generally. There’s a PCMI pset that starts with that, and I’ve done it with 5th, 6th, 10th and 11th graders. (Admittedly, the 11th graders didn’t love it. But everyone else loved it.)