One more of those

S=n=1σ2(n)n6S=\sum_{n=1}^{\infty}\frac{\sigma_2(n)}{n^6}

Let σ2(n)\sigma_2(n) denote the sum of the squares of all the positive integer divisors of nn. For example, σ2(6)=12+22+32+62=50\sigma_2(6)=1^2+2^2+3^2+6^2=50.

Enter π10S\frac{\pi^{10}}{S} as your answer.

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