\[S=\sum_{n=1}^{\infty}\frac{\sigma_2(n)}{n^6}\]

Let \(\sigma_2(n)\) denote the sum of the squares of all the positive integer divisors of \(n\). For example, \(\sigma_2(6)=1^2+2^2+3^2+6^2=50\).

Enter \(\frac{\pi^{10}}{S}\) as your answer.

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