Let $a_1,a_2,\dots$ be a sequence of positive integers such that $m+n$ divides $a_m+a_n$ for all $m<n$. Is it necessary that $n$ divides $a_n$ for all $n$?
Examples of such a sequence is $a_n=kn$ for some positive integer $k$. From $m+n$ dividing $a_m+a_n$ we see that if we fix $a_1,\dots,a_{n-1}$, then $a_n$ is fixed modulo $n+1,n+2,\dots,2n-1$.
Following the hints in the comments:
$$4n\mid a_n+a_{3n}, 6n\mid a_n+a_{5n}, 8n\mid a_{3n}+a_{5n}$$
So $2n\mid (a_n+a_{3n})+(a_n+a_{5n})-(a_{3n}+a_{5n})=2a_n$, hence $n\mid a_n$.