Generalizing Ramanujan's "1729 story"

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 $2$.

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.

• 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