Divisor Sigma Function

Calculus Level 5


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

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

Find \(\lceil 10000S \rceil\)


  • \(\tau (x)\) 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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