For each positive integer
![k,](/media/m/3/b/3/3b3f59143c4e876b4adc54b666df1462.png)
let
![t(k)](/media/m/1/1/b/11b46aff1180121ea1e4c2b41ed31a28.png)
be the largest odd divisor of
![k.](/media/m/3/5/d/35d6ded5ad555e4f371e82635560eb35.png)
Determine all positive integers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
for which there exists a positive integer
![n,](/media/m/5/f/2/5f26ebab144fee216fbc733cb1fa2f2b.png)
such that all the differences
![t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)](/media/m/6/8/c/68cb3fa02376bcf8a2654b7ce13482dc.png)
are divisible by 4.
Proposed by Gerhard Wöginger, Austria
%V0
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
$$t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)$$ are divisible by 4.
Proposed by Gerhard Wöginger, Austria