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Let $n$ be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute $n$.

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A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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Official solution

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

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.