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In this paper, we prove the following result: {quote} Let $\A$ be an infiniteset of positive integers. For all positive integer $n$, let $\tau_n$ denote thesmallest 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 caseswhen $\A$ is the set of all positive integers and when $\A$ is the set of theprime numbers, we give a more precise result for the average asymptoticbehavior of ${(\tau_n)}_n$. Furthermore, we discuss the irrationality of thelimit of $\tau_n$ (in the average sense) by applying a result of Erd\H{o}s.

On the series of the reciprocals lcm's of sequences of positive integers: A curious interpretation