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For a nonnegative integer n, define a_n to be the positive integer with decimal representation
1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}
Prove that \frac{a_n}{3} is always the sum of two positive perfect cubes but never the sum of two perfect squares.

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