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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