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Problem 834

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Add and Divide

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 $21,33,46,60,75,91,108,126,\ldots$.

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 $U(1234567)$.

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相加与整除

构造如下数列：选择任意正整数$n$，在第$m$步时给当前的整数增加$(n+m)$并作为数列的第$m$项。例如，若$n=10$，则构造的数列为$21,33,46,60,75,91,108,126,\ldots$

集合$S(n)$包含所有满足下列条件的下标$m$：数列的第$m$项能够被$(n+m)$整除。

例如，$S(10)={5,8,20,35,80}$。

记$T(n)$为$S(n)$中所有下标之和。例如，$T(10) = 148$，$T(10^2)=21828$。

记$\displaystyle U(N)=\sum_{n=3}^{N}T(n)$。已知$U(10^2)=612572$。

求$U(1234567)$。

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