  In this paper, we prove the following result: {quote} Let $\A$ be an infinite
set of positive integers. For all positive integer $n$, let $\tau_n$ denote the
smallest element of $\A$ which does not divide $n$. Then we have $$\lim_{N \to
+ \infty} \frac{1}{N} \sum_{n = 1}^{N} \tau_n = \sum_{n = 0}^{\infty}
\frac{1}{\lcm\{a \in \A | a \leq n\}} .$${quote} In the two particular cases
when $\A$ is the set of all positive integers and when $\A$ is the set of the
prime numbers, we give a more precise result for the average asymptotic
behavior of ${(\tau_n)}_n$. Furthermore, we discuss the irrationality of the
limit of $\tau_n$ (in the average sense) by applying a result of Erd\H{o}s.
