Hammingtons Mayhem

Mr. Hammington really likes $1$. He gives you a number $N$ in base $2$. Let $H(N)$ denote the number of set bits(i.e. $1$s) in $N$. Let $H^n(N)$ is defined as function obtained by applying function $H(\cdot)$, $n-$times to $N$. Formally, - $H^1(N)=H(N)$ - $H^n(N)=H(H^{n-1}(N))$ for $n\geq2$. Mr. Hammington describes a special function $F(N)$, which denotes the smallest value of $n$ such that $H^