I have thought about problem 386 for 2 months and I have given up. A divisor antichain of a number is a subset of its (positive) divisors no one of which is divisible by another (e.g. for $30$ $(2,3,5)$ is an antichain). The problem is to find the sum of the length of all maximal length divisor antichains of numbers from $2$ to $10^8$ (for $30$ it is easy to see that it is $3$). For individual numbers the maximal length can be calculated (given we know the number of distinct prime factors and their degree (e.g. $60$ is of the form $p^2qr$ where $p,q,r$ are $2,3,5$)) .
(Given that, I guess, the acceptable number of unit tasks in a ProjectEuler problems is about $10^7$ so even if we knew the factorisations of all numbers up to $10^8$ the algorithm would be too slow. Another problem is that I don't see how we can avoid knowing all primes up to $10^8$ (their list would take up too much memory space) and solve it.)
I tried to connect the problem to Dilworth's theorem but that didn't help.
Can you give me a hint? I won't submit the solution now and I don't want to copy-paste the answer to unlock forum discussions because so few people have solved it (every result of Google for this problem seems to be running too slowly).
The length of the divisor antichain for squarefree numbers is given by Sperner's theorem.