关于Smarandache函数的一个问题

On a problem of the Smarandache function

对任意正整数n,著名的Smarandache函数S(n)定义为最小的正整数m使得n|m!.即就是S(n)=min{m:m∈N,n|m!}.令PS(n)表示区间[1,n]叫中S(n)为素数的正整数n的个数.在一篇未发表的文献中,J.Castillo建议我们研究当n→∞时,比值PS(n)/n的极限存在问题.如果存在,确定其极限.本文的主要目的是利用初等方法研究这一问题,并得到彻底解决!即就是证明该极限存在且为1.

For any positive integer n, the famous Smarandache function S（n） is defined as the smallest positive integer m such that nlm!. That is,S（n）=min{m：m∈N,n｜m!}.Let PS（n） denotes the number of all n in the interval [1, n] such that S（n） be a prime. In an unpublished paper, J.Castillo asked us to determine the limit PS（n）/n as n → ∞, if this limit exists, find its value. In this paper, we using the elementary method to study this problem, and prove that its limit exists, and its value is 1.