Problem 563
Robot Welders
A company specialises in producing large rectangular metal sheets, starting from unit square metal plates. The welding is performed by a range of robots of increasing size. Unfortunately, the programming options of these robots are rather limited. Each one can only process up to 25 identical rectangles of metal, which they can weld along either edge to produce a larger rectangle. The only programmable variables are the number of rectangles to be processed (up to and including 25), and whether to weld the long or short edge.
For example, the first robot could be programmed to weld together 11 raw unit square plates to make a 11×1 strip. The next could take 10 of these 11×1 strips, and weld them either to make a longer 110×1 strip, or a 11×10 rectangle. Many, but not all, possible dimensions of metal sheets can be constructed in this way.
One regular customer has a particularly unusual order. He always demands that the finished product should have an exact area, and that the long side must not be more than 10% larger than the short side. If these requirements can be met in more than one way, in terms of the exact dimensions of the two sides, then he demands that all variants are produced. For example, if he were to ask for metal sheet of area 889200, then there are three final dimensions that can be produced: 900×988, 912×975 and 936×950. The target area of 889200 is the smallest area which can be manufactured in three different variants, within the limitations of the robot welders.
Let M(n) be the minimal area that can be manufactured in exactly n variants with the longer edge not greater than 10% bigger than the shorter edge. Hence M(3) = 889200.
Find $\sum_{n=2}^{100} M(n)$.
焊接机器人
有一家专门将单位方形钢板焊接成大长方形钢板的公司，其中所有的焊接操作都由一系列尺寸递增的机器人完成。不幸的是，这些机器人的设置选项很有限，每一台机器人只能同时处理至多25块同样大小的长方形钢板，将其按照长边或短边焊接成一个更大的长方形，这其中可以进行设置的变量只有同时处理的钢板数量（至多25块）和按照长边还是短边进行焊接。
例如，第一台机器人可以设置为同时处理11块单位正方形钢板原材料，将其焊接为一个11×1的长条。第二台机器人可以将10条11×1的长条焊接成一个更长的110×1的长条，或者是一个11×10的长方形。通过这种方式可以构造出许多种钢板，但是并不是每一种钢板都能制造出来。
一位常客的订单总是特别不同寻常。他总是要求最终产品的面积是一个由他给定的值，而且长边不能比短边长超过10%。如果有多种最终产品能够满足他的要求，他希望每一种都被生产出来。例如，如果他想要面积是889200的钢板，一共有三种最终产品符合要求的钢板会被生产出来：900×988，912×975和936×950。在焊接机器人本身的限制下，889200这一目标面积是使得最终产品有三种可能的最小面积。
记M(n)是使得可制造的、长边比短边长不超过10%的最终产品有恰好n种可能的最小面积，已知M(3) = 889200。
求$\sum_{n=2}^{100} M(n)$。