HMMT 2021 T6

Let $f(x)=x^{2}+x+1$ . Determine, with proof, all positive integers $n$ such that $f(k)$ divides $f(n)$ whenever $k$ is a positive divisor of $n$ .

Dataset text NuminaMath olympiads. Quoted with source.

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n = 1, \text{ a prime that is } 1 \bmod 3, \text{ or the square of any prime except 3}

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