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 $\sum N(n)$ for $1 \le n \le 10^8$.

The presented task is extremely complex and involves a significant degree of computation, particularly for a human tutor. Computing $N(n)$ for $1 \le n \le 10^8$ is a problem that requires complex algorithms and efficient computational resources. This problem isn’t feasible by manual calculations due to the size of the dataset.

However, to approach this problem programmatically, you could use the following strategy:

1. Observe that a factor of $n$ can be written in a prime factorization form, where the prime factors come from the prime factorization of $n$. For instance, the prime factorization of $30$ is $2\times 3\times 5$. Hence any factor of $30$ can be expressed as $2^x \times 3^y \times 5^z$, where $x,y,z \in \{0,1\}$.
2. Next, consider the set of all possible values for $(x,y,z)$ for a given $n$, and form a directed graph where each node represents a unique combination of $(x,y,z)$, and there is an edge from node $A=(a1,a2,a3)$ to node $B=(b1,b2,b3)$ if and only if $a1<=b1, a2<=b2$ and $a3<=b3$. The graph would then represent the divisibility relation among the factors of $n$. 3. To find $N(n)$, the maximum length of an antichain of $S(n)$, we want to find the largest set of nodes in the graph such that there is no path from any node in the set to any other node in the set. This is known as finding the maximum independent set of the graph, which is a well-known problem in computer science. 4. Implement this algorithm and run it for $n=1$ to $n=10^8$, and sum up the results. This is a geometric approach to the problem based on the observation that the problem is essentially about lattice paths in three dimensions. It should be mentioned again that this problem is not feasible for hand calculations and requires programming skills and computational resources. I would suggest to use a programming language such as Python or C++, which are frequently used for complex mathematical calculations and problem solving techniques.

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