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Problem 386

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Maximum length of an antichain

Let n be an integer and S(n) be the set of factors of n.

A subset A of S(n) is called an antichain of S(n) if A contains only one element or if none of the elements of A divides any of the other elements of A.

For example: S(30) = {1, 2, 3, 5, 6, 10, 15, 30}
{2, 5, 6} is not an antichain of S(30).
{2, 3, 5} is an antichain of S(30).

Let N(n) be the maximum length of an antichain of S(n).

Find ΣN(n) for 1 ≤ n ≤ 10⁸

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反链的最大长度

取整数n，记S(n)为n的约数构成的集合。

若S(n)的子集A只包含一个元素，或者A中任意一个元素均不能整除其它元素，则称A为S(n)的反链。

例如：S(30) = {1, 2, 3, 5, 6, 10, 15, 30}
{2, 5, 6}不是S(30)的反链。
{2, 3, 5}是S(30)的反链。

记N(n)为S(n)的反链的最大长度。

对于1 ≤ n ≤ 10⁸，求ΣN(n)。

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