Tallying Tiny Logs

Algebra Level 2

S=2×2016log2(2016!2)+3×2016log3(2016!3)+4×2016log4(2016!4)++2016×2016log2016(2016!2016)S=\dfrac{2\times2016}{\log_{2}(2016!^{2})}+\dfrac{3\times2016}{\log_{3}(2016!^{3})}+\dfrac{4\times2016}{\log_{4}(2016!^{4})}+\dots+\dfrac{2016\times2016}{\log_{2016}(2016!^{2016})}

Find SS.

Notation: !! denotes the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

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