A sequence is created by starting with a positive integer $n$ and incrementing by $(n+m)$ at the $m^{th}$ step. If $n=10$, the resulting sequence will be $21,33,46,60,75,91,108,126,\dots$.

Let $S(n)$ be the set of indices $m$, for which the $m^{th}$ term in the sequence is divisible by $(n+m)$.
For example, $S(10)=\{5,8,20,35,80\}$.

Define $T(n)$ to be the sum of the indices in $S(n)$. For example, $T(10)=148$ and $T({10}^2)=21828$.

Let ${\displaystyle U(N)=\sum _{n=3}^{N}T(n)}$. You are given, $U({10}^2)=612572$.

Find $U(1234567)$.