Slightly odd

\[\large S_n=\sum_{i=1}^n \left \lfloor{\frac{n}{i}}\right \rfloor\]

\(S_n\) is defined as of above, for all positive integers \(n\). Determine the number of positive integers \(1<a<1616\) for which the value of the expression

\[\large S_{a-1}+S_{a}\]

is an odd positive integer.

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