The number 1729 is famously the smallest positive integer expressible as the sum of two positive cubes in two different ways ($1729=1^3+12^3=9^3+10^3$). There is plenty of work on "taxicab numbers" - the smallest sums of cubes in $n$ different ways (which always exist) - Here's Ivars Peterson at MAA And here's another detailed analysis. (Does anyone know anything about the "Bill Butler" referred to in the second article)
However the sequence which caught my attention is OEIS A016078 - 4, 50, 1729, 635318657 which gives the smallest numbers which are sums of positive $n^{th}$ powers in two ways. Is there any more recent work or prospect of identifying such numbers for fifth powers and above? And should they be named as in the title of this post?
[This question arises from a much more frivolous one, which was closed, in which I learned why a $50^{th}$ birthday was special in this particular way].