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Let $H_n = 1+\frac12 + \frac13 + \cdots + \frac1{n}$ be the $n^\text{th}$ harmonic number.

The sum $\displaystyle \sum_{n=1}^\infty \frac{H_n}{2^n} = \ln (a)$ for some positive integer $a.$

What is $a?$

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