Here is a rather nice calculator, now donated to the Science Museum in London.
Cyprian’s Last Theorem
Cyprian’s Last Theorem (henceforth CLT) states that the sum of n consecutive nth powers is never equal to the next nth power in the sequence, unless n=2 (32+42=52) or n=3 (33+43+53=63). I haven't proved it yet but I'm having fun trying.
How many orders are there?
In how many ways can you put n items into sequence? If every item is distinct, then the answer is n! - for instance, for 3 items the answer is 6:
But if some items are allowed to be equal, the number is 13: