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Problem 183

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Maximum product of parts

Let N be a positive integer and let N be split into k equal parts, r = N/k, so that N = r + r + … + r.
Let P be the product of these parts, P = r × r × … × r = r^k.

For example, if 11 is split into five equal parts, 11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2, then P = 2.2⁵ = 51.53632.

Let M(N) = P_max for a given value of N.

It turns out that the maximum for N = 11 is found by splitting eleven into four equal parts which leads to P_max = (11/4)⁴; that is, M(11) = 14641/256 = 57.19140625, which is a terminating decimal.

However, for N = 8 the maximum is achieved by splitting it into three equal parts, so M(8) = 512/27, which is a non-terminating decimal.

Let D(N) = N if M(N) is a non-terminating decimal and D(N) = -N if M(N) is a terminating decimal.

For example, ΣD(N) for 5 ≤ N ≤ 100 is 2438.

Find ΣD(N) for 5 ≤ N ≤ 10000.

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最大拆分乘积

N是一个正整数，将N拆分成k个相等的部分，记r = N/k，也即N = r + r + … + r。
记P是这些拆分的乘积，也即P = r × r × … × r = r^k。

例如，将11拆分成5个相等的部分，11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2，那么P = 2.2⁵ = 51.53632。

对于给定的N，记M(N) = P_max。

计算可得，N = 11时，这个最大值是在将其拆分成四等份时得到的，此时P_max = (11/4)⁴，也就是说M(11) = 14641/256 = 57.19140625，这是一个有限小数。

然而，对于N = 8，这个最大值是在将其拆分成三等份时得到的，此时M(8) = 512/27，不是有限小数。

若M(N)不是有限小数，记D(N) = N，反之记D(N) = -N。

例如，对于5 ≤ N ≤ 100，ΣD(N) 为2438。

对于5 ≤ N ≤ 10000，求ΣD(N)。

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