And again the chain

Consider a chain consisting of $N$ rings. Determine the minimum number of rings that must be unlinked so that the remaining segments can be rearranged to form chains of any length from $1$ to $N$ rings. The unlinked rings can be reused when forming new chains.

For example, if N = 21, it is enough to unlink just $2$ rings to create segments of lengths $3$, $5$, and $11$. The two unlinked rings are considered as chains of length one.

Write a program that, given a natural number $N$ (the length of the initial chain), calculates the minimum number of rings that must be unlinked to achieve the described goal.