Pathetic numbers.

Number Theory Level 5

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


are divisible by \(4\).

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


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