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 $1≤a≤13^{9}$ .