HMMT November 2015 U34

Let $n$ be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute $n$. If your guess is $a$, you will receive $\max \left(25-5 \cdot \max \left(\frac{a}{n}, \frac{n}{a}\right), 0\right)$ points, rounded up.