Generalizing Ramanujan's "1729 story" Whenever I read the anecdote about Hardy, Ramanujan and the taxi number 1729 I'm amazed that it could have occurred to anyone just off the top of their head that 1729 can be written as the sum of two cubes in two different ways -- and that it is the smallest such number.
At all events, there are several ways to look at this in a more general way. For positive integers $n, k$ let us set $$r_n(k) = | \{(x,y): x\leq y \text{ and } x^n + y^n = k\} |.$$
For what, if any, $n,y\geq 2$ is $r_n^{-1}(\{y\})$ is infinite? 
(Also partial answers and/or examples are very welcome.)
 A: There are many articles that study the quantity you call $r_n(k)$ using sieve methods. Among them I mention the following, which give highly non-trivial bounds for the number of $k<X$ such that $r_n(k)>1$. If you look at these, and also forward reference them using MathSciNet, you should be able to find the state of the art.


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*T.D. Browning, Equal Sums of Two kth Powers, Journal of Number Theory, Volume 96, Issue 2, October 2002, Pages 293–318.

*C. Hooley. On another sieve method and the numbers that are a sum of two hth powers. Proc. London Math. Soc., 226 (1981), pp. 30–87.

*C. Hooley, On another sieve method and the numbers that are a sum of two hth powers: II. Journal für die reine und angewandte Mathematik (1996) Volume: 475, page 55-76    

A: You may be interested in this article The 1729 $K3$ surface by Ken Ono and Sarah Trebat-Leder, which is aimed at exactly this question of how Ramanujan knew $1729$ so well.  Briefly, Ramanujan had studied parameterizations of $a^3+b^3= c^3+ d^3$ in detail, and especially the near misses to Fermat for cubes $a^3 + b^3 =c^3 \pm 1$.  The paper by Ono and Trebat-Leder sets this in the context of demonstrating that a certain elliptic curve over ${\Bbb Q}(t)$ has rank $2$.  
