Abstract
Let $f$ be an arithmetical function. A set $S=\{x_1, \dots , x_n\}$ of $n$ distinct positive integers is called multiple closed if $y\in S$ whenever $x\,|\, y\,|\,{\rm lcm}(S)$ for any $x\in S$, where ${\rm lcm}(S)$ is the least common multiple of all ele
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