A word $w$ on the alphabet $A := \{0, 1\}$ is *factorable* if
\begin{equation}
 w = u^k \mbox{ where } u \in A^\* \mbox{ and } k \geq 2.
\end{equation}


Let $L$ be the language of the set of factorable words on $A$ and $f(t)$ be its generating series, that is
\begin{equation}
 f(t) := \sum\_{n \geq 0} |L \cap A^n| \; t^n.
\end{equation}



> 
> What is the generating function of $L$?
> 
> 
> 



Generating function of factorable binary words
The ordinary generating function is a fairly horrible Lambert expansion (usually with a natural boundary). For more on this subject look in my old paper *[Walks on Free Groups and other Stories](http://arxiv.org/abs/1106.5947)*.


