Yet another footnote to the least non zero digit of \(n!\) in base 12. (English) Zbl 1454.11018

Let \(\ell_b(n!)\) denote the least non-zero digit of \(n!\) in a given base \(b\). In G. P. Dresden’s proof [Math. Mag. 81, No. 2, 96–105 (2008; Zbl 1165.11060)] that the number \(\sum_n \ell_{10}(n!)10^{-n}\) is transcendental, the fact that the sequence \(\mathcal{L}=\{\ell_{10}(n!)\}_n\) is 5-automatic, but not periodic, was instrumental. The case \(b=12\) is the smallest one when this condition cannot be guaranteed. J.-M. Deshouillers and I. Z. Ruzsa [Publ. Math. 79, No. 3-4, 395–400 (2011; Zbl 1249.11044)] proved that the set \(\mathcal{L}_a=\{n : \ell_{12}(n!)=a\}\), \(0\le a\le 11\), has asymptotic density \(1/2\) if \(a=4\) or \(a=8\) and \(0\) otherwise. This results raised the question whether for the 12-ary digits \(a\neq 4,8\) these sets are infinite. The question was answered in affirmative if \(a\in\{3,6,9\}\) in the author’s paper [Unif. Distrib. Theory 7, No. 1, 71–73 (2012; Zbl 1313.11024)]. Despite the fact that \(\mathcal{L}\) coincides on a set of asymptotic density 1 with a 3-automatic sequence taking only the values 4 and 8, each with asymptotic density \(1/2\), in the paper under review the author proves that the sets \(\mathcal{L}_4\) and \(\mathcal{L}_8\) are not 3-automatic and that the sets \(\mathcal{L}_a\) with \(a=3,6,9\) are not automatic.
The author closes this paper with the remark that the non automaticity of \(\mathcal{L}_4\) and \(\mathcal{L}_8\) in all bases is still open.


11A63 Radix representation; digital problems
11B85 Automata sequences
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