Problem 834

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

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

Find $U (1234567)$ .

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 $21, 33, 46, 60, 75, 91, 108, 126, \dots$

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

求 $U (1234567)$ 。