Divisor Sigma Function

Calculus Level 5

If

$S=\displaystyle \sum_{p} \displaystyle \sum_{n=2}^{\infty} \dfrac{\tau (n)-1}{p^n}$

where $$p\in\mathbb{N}$$ is greater than 1 and is NOT a perfect power.

Find $$\lceil 10000S \rceil$$

Notes:

• $$\tau (x)$$ is the divisor sigma function. This represents the number of factors of a number including 1 and itself. For example, $$\tau(6)=\sigma_0(6)=4$$.

• $$\lceil \cdot \rceil$$ denotes the ceiling function.

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