Journal of Integer Sequences, Vol. 14 (2011), Article 11.6.8

On the Base-b Expansion of the Number of Trailing Zeros of bk !

Antonio M. Oller-Marcén
Departamento de Matemáticas
Universidad de Zaragoza
C/Pedro Cerbuna 12
50009 Zaragoza

José María Grau
Departamento de Matematicas
Universidad de Oviedo
Avda. Calvo Sotelo, s/n,
33007 Oviedo


Let $Z_{b}(n)$ denote the number of trailing zeroes in the base-$b$ expansion of $n!$. In this paper we study the connection between the expression of $\vartheta(b):=\lim_{n\rightarrow \infty}Z_{b}(n)/n$ in base $b$, and that of $Z_{b}(b^{k})$.

In particular, if $b$ is a prime power, we will show the equality between the $k$ digits of $Z_{b}(b^{k})$ and the first $k$ digits in the fractional part of $\vartheta (b)$. In the general case we will see that this equality still holds except for, at most, the last $\left\lfloor \log _{b}(k)\ +3\right\rfloor $ digits. We finally show that this bound can be improved if $b$ is square-free and present some conjectures about this bound.

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(Concerned with sequences A000966 A011371 A027868 A054861 A090624 A173228 A173292 A173293 A173345 A173558 A174807 A181582.)

Received February 25 2011; revised version received June 7 2011. Published in Journal of Integer Sequences, June 10 2011.

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