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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