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Let p=1+1+1+…p = \sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}p=1+1+1+….
The sum
∑k=2∞⌊pk⌉2k\sum_{k=2}^{\infty} \frac{\lfloor p^{k} \rceil}{2^{k}}k=2∑∞2k⌊pk⌉
can be expressed as ab\frac{a}{b}ba for a,ba,ba,b coprime, where ⌊⋅⌉\lfloor \cdot \rceil⌊⋅⌉ denotes the nearest integer function. Find a+ba+ba+b.
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