Automaticity of the sequence of the last nonzero digits of \(n!\) in a fixed base. (English. French summary) Zbl 1475.11039

Let \(b \ge 2\) be an integer. A sequence \((a_n)_{n \ge 0}\) taking finitely many values is said to be \(b\)-automatic if there is a finite machine which permits to know the value of \(a_n\) by reading one after the other the digits of \(n\) in its expansion in base \(b\). Moreover, the sequence \((a_n)_n\) is said to be automatic if it is \(k\)-automatic for some \(k\). This classical notion is recalled in Section 2 as well as its main properties – a useful point.
The author addresses the question of the automaticity of the sequence \((\ell_b(n!))_n\), where \(\ell_b(n!)\) - an integer in \([1, b-1]\) - denotes the last nonzero digit of \(n!\) in base \(b\), defined in the following way: if one writes \(n! = b^r m\), where \(\gcd(m, b) \neq 0\), then \(\ell_b(n!) \) is congruent to \(\gcd(m,b)\) modulo \(b\).
It is folklore that \((\ell_b(n!))_n\) is automatic if \(b\) is a power of a prime, or a small number like \(6\) or \(10\). It was proved in the paper quoted as [4] that the sequence \((\ell_{12}(n!))_n\) is not automatic.
The paper completely characterizes the values of \(b\) for which the sequence \((\ell_b(n!))_n\) is automatic. More precisely, Theorem 3, proved in Section 3, states the following
Let \(b = p_1^{a_1} p_2^{a_2}\cdots\) with \(a_1(p_1-1) \ge a_2(p_2-1)\ge \cdots\). The sequence \((\ell_b(n!))_n\) is \(p_1\)-automatic if \(a_1(p_1-1) > a_2(p_2-1)\) or \(b = p_1^{{a_1}}\) and is not automatic otherwise.
Notice that \(12=2^2 \times 3^1\) with \(2\times (2-1) = 1 \times (3-1)\).


11B85 Automata sequences
11A63 Radix representation; digital problems
68Q45 Formal languages and automata
68R15 Combinatorics on words
Full Text: DOI arXiv


[1] Allouche, Jean-Paul; Shallit, Jeffrey, Automatic Sequences. Theory, Applications, Generalizations (2003), Cambridge University Press · Zbl 1086.11015
[2] Byszewski, Jakub; Konieczny, Jakub, A density version of Cobham’s theorem (2017) · Zbl 1477.11049
[3] Deshouillers, Jean-Marc, A footnote to The least non zero digit of \(n!\) in base 12, Unif. Distrib. Theory, 7, 1, 71-73 (2012) · Zbl 1313.11024
[4] Deshouillers, Jean-Marc, Yet another footnote to The least non zero digit of \(n!\) in base 12, Unif. Distrib. Theory, 11, 2, 163-167 (2016) · Zbl 1454.11018
[5] Deshouillers, Jean-Marc; Ruzsa, Imre, The least non zero digit of \(n!\) in base 12, Publ. Math., 79, 3-4, 395-400 (2011) · Zbl 1249.11044
[6] Legendre, Adrien-Marie, Théorie des nombres (1830), Firmin Didot frères · Zbl 1395.11005
[7] Stewart, Cameron L., On the representation of an integer in two different bases, J. Reine Angew. Math., 319, 63-72 (1980) · Zbl 0426.10008
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