# On the least non-zero digit of $n!$

Gregory P. Dresden has studied the sequence $(\ell_{10}(n!))$, where $\ell_b(m)$ denotes the least non zero digit of $m$ written in the base $b$ (e.g., $\ell_{10}(4035000)=5$). He showed that this sequence is 5-automatic, but he did not express it in those terms. Imre Ruzsa has studied more generally the sequence $(\ell_{b}(n!))$, which is not necessarily an automatic sequence. We shall illustrate our result by concentrating on the case when $b=12$. We shall prove that, although the sequence $(\ell_{12}(n!))$ may not be automatic, each digit $d\in\{1,\ldots, 11\}$ appears in the sequence $(\ell_{12}(n!))$ with a natural density. (This is joint work with Imre Ruzsa.)

Jean-Marc Deshouillers