Lambert series

$\sum_{p} \frac{1}{2^p - 1} = \sum_{p} \frac{2^{-p}}{1 - 2^{-p}} = \sum_{p} \sum_{n \geq 0} 2^{-np - p} = \sum_{p} \sum_{n \geq 1} 2^{-np} = \sum_{k \geq 1} \left( \sum_{p | k} 1 \right) 2^{-k}$

The first equality follows from algebraic manipulation, second equality follows from the power series of $\frac{1}{1 - x}$ , third equality follows by a transformation on the indices of summation and final equality because the term $x^{-k}$ will appear as many times as there are primes that divide $k$ .

$\sum_{p | 10^{14}} 1 = 1 + 1 = 2$ as the only prime divisors of $10^{14}$ are 2 and 5.