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arXiv · 1907.00370

A proof of Sondow's conjecture on the Smarandache function

Xiumei Li·Min Sha
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Abstract

The Smarandache function of a positive integer $n$, denoted by $S(n)$, is defined to be the smallest positive integer $j$ such that $n$ divides the factorial $j!$. In this note, we prove that for any fixed number $k > 1$, the inequality $n^k < S(n)!$ holds for almost all positive integers $n$. This confirms Sondow's conjecture which asserts that the inequality $n^2 < S(n)!$ holds for almost all positive integers $n$.

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Xiumei Li, Min Sha. 2019-06-30. A proof of Sondow's conjecture on the Smarandache function. https://arxiv.org/abs/1907.00370

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