The naturals $\mathbb{N}$ can be encoded with a binary unary code such as
$$ \begin{align} 1&=0_b\\ 2&=10_b\\ 3&=110_b\\ &... \end{align} $$ The length this the encoding grows linearly with the natural number it represents. For example the length of $110_b$ is 3 and it also represents the number 3.
I am looking for an upper bound on the optimality of the prefix free encoding. For example, can we design a prefix coding of $\mathbb{N}$ such that the length of the encoding grows according to ~$\sqrt{n}$, instead of $n$. Is there an upper bound on the optimality that we can prove?