Any solutions

The condition means that $i+j\mid i(a_i-a_j)$ and since $(k,2k-1)=1$ we get $2k-1 \mid a_k - a_{k-1}$. So if $2k-1>2016$, then $a_k=a_{k-1}$, since $0<a_k, a_{k-1}\leq 2016$, therefore $a_{1009}=a_{1010}=\ldots = a_{2016}$. And if $i<1009$, then $(i,j)=1$ for some $j\in\{ 2017-i, 2017-i+1, \ldots , 2016\}$, so $i+j\mid a_i-a_j$ and similarly because $i+j>2016$ we have $a_i=a_j=a_{2016}$, therefore all such sequences are constant.