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