
Here is a rather nice calculator, now donated to the Science Museum in London. Cyprian’s Last TheoremCyprian’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:
This "number of weak orders on n labelled elements" is number A000670 in Sloane's OnLine Encyclopaedia of Integer Sequences. Here is a paper (PDF) investigating its properties. 