HMMT-Nov 2015 guts, Problem 34

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.

Solution: A computer search yields that the second smallest number is 4104. Indeed, $4104=9^{3}+15^{3}=2^{3}+16^{3}$ $\fbox{4104}$.

Answer: 4104