Pathetic numbers.

For a positive integer \(k\), let \(t(k)\) be the largest odd divisor of \(k\).

A positive integer \(a\) is said to be pathetic if there exist a positive integer \(n\) such that all the differences

\(t(n+a)-t(n),t(n+a+1)-t(n+1),....,t(n+2a-1)-t(n+a-1)\)

are divisible by \(4\).

Find the sum of all pathetic numbers \(a\) such that \(1≤a≤13^{9}\).

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