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\displaystyle\sum_{p\in\mathmm{P}}^{\infty}\frac{1}{p}
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s = \O
\begin{aligned} \displaystyle \sum_{s \in \mathcal{P}(S_{n+1}), s \not= \O} \frac{\sigma(s)}{\pi(s)} & = \sum_{s \in \mathcal{P}(S_{n}), s \not= \O} \frac{\sigma(s)}{\pi(s)} + \sum_{s \in \mathcal{P}(S_{n} )} \frac{n+1 + \sigma(s)}{(n+1)\pi(s)} \\&= n(n+2) - (n+1)H_{n} + 1+n + \frac{1}{n+1}(n(n+2) - (n+1)H_{n}) \\& = n(n+2) - (n+1)H_{n} + 2n+2 - \frac{1}{n+1} -H_{n} \\&= (n+1)(n+3) - (n+2)H_{n+1} \end{aligned}
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