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:

1. a>b>c
2. a>c>b
3. b>a>c
4. b>c>a
5. c>a>b
6. c>b>a

But if some items are allowed to be equal, the number is 13:

7. a=b>c, which is the same as b=a>c.
8. a=c>b, or c=a>b.
9. b=c>a, or c=b>a.
10. a>b=c, or a>c=b.
11. b>a=c, or b>c=a.
12. c>a=b, or c>b=a.
13. a=b=c (or b=c=a, etc).

This "number of weak orders on n labelled elements" is number A000670 in Sloane's On-Line Encyclopaedia of Integer Sequences. Here is a paper (PDF) investigating its properties.