## Maximum length of an antichain

### Problem 386

Published on Sunday, 27th May 2012, 08:00 am; Solved by 403; Difficulty rating: 40%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 Σ`N`(`n`) for 1 ≤ `n` ≤ 10^{8}