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.

中文翻译：

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关于Smarandache函数的Sondow猜想的证明

摘要

将由S（n）表示的正整数n的Smarandache函数定义为最小的正整数j，以使n除以阶乘$Ĵ！$。在本说明中，我们证明了对于任何固定数k  > 1，不等式${ñ}^{ķ}<\mathrm{小号}（ñ）！$几乎适用于所有正整数n。这证实了桑多的猜想，后者断定不平等${ñ}^2<\mathrm{小号}（ñ）！$几乎适用于所有正整数n。