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 \(a\)?


This problem is posed by Michael T.
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