# Michael's double factorial

For positive integers $$n,$$ the double factorial function is defined as $n!! = \begin{cases} n(n-2)(n-4)\ldots\cdot4\cdot2, \; n \; \text{even} \\ n(n-2)(n-4)\ldots\cdot3\cdot1, \; n \; \text{odd} \end{cases}$ The ratio $$\dfrac{513!}{513!!}$$ can be written as $$2^a \cdot b!$$ for positive integers $$a$$ and $$b$$ in several different ways. For the smallest value of $$b$$, what is the corresponding value of $$a$$?

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