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Number Theory Level 5

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

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.