There are $n$ evenly spaced nails on a wall, parallel to the floor. Saketh wants to hang a picture on the nails so that when all the nails are in place, the picture doesn’t fall, but when Sreenath removes any one nail, the picture does fall. The infinitely strong, infinitely thin, weightless wire of arbitrarily long length is already attached to the picture frame. Let $f(n)$ be the minimum number of times Saketh can do clockwise and counterclockwise wraps around the nails to set up this system (note: $f(1) = 1$). Find: $$\sum\limits_{i=1}^{32} f(i)$$