Prime Telescoping Series

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We know that $$ \log{2} = \sum_{n=1}^{\infty} {\frac{1}{2n (2n-1)}} \space < \space \sum_{p \space prime} {\frac{1}{p (p-1)}} \space < \space \sum_{n=2}^{\infty} {\frac{1}{n (n-1)}} = 1 $$ Is there any formula for the mid sum? (constant, value, limit, integral, function)