Question 3
For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$.
Determine all positive integers $a$ for which there exists 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 $4$.
Determine all positive integers $a$ for which there exists 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 $4$.