Smarandache function

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Given a natural number $n$, the value of the Smarandache function $ \eta $ at $n$ is the smallest natural number $m$ such that $n$ divides $m!$. An elementary observation is that $\eta ( n ) \leq n$, and that $\eta ( n ) = n$ if and only if $n$ is a prime number or equal to $4$.


[a1] F. Smarandache, "A function in number theory" Smarandache Function J. , 1 (1990) pp. 3–65
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Smarandache function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article