Pebrudal's logarithmic sum

Let \(a\) and \(b\) be positive integers such that

\[\sum_{n=4}^{2^{200}-1}\left \lfloor \log_{2} n \right \rfloor=a\cdot 2^{b},\]

where \(a\) is an odd integer and \(b\) is a positive integer. What is \(a+b\)?

This problem is posed by Pebrudal Z.

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