一个关于Smarandache LCM对偶函数的方程

An equation involving the Smarandache LCM dual function

n∈N+,著名的Smarandache LCM函数的对偶函数定义为SL*(n)=max{k|[1,2,…,k]|n,k∈N+},Ω(n)表示n的所有素因子的个数.利用初等数论和分类讨论的方法研究了一个包含SL*(n)及素因子函数方程∑d|n1SL*(d)=Ω(n)的可解性,并给出了这个方程的所有正整数解的具体形式.

For any positive integer n ,the well-know Smarandache LCM dual function was defined as SL＊（n）= max{k｜ [1 ,2 ,？ ,k]｜ n ,k ∈ N＋} ,Ω（n）was the number of all the prime factors of n .By using the elementary number theory and classification discussion methods to study the solvability of the equa-tion ∑d｜n 1SL ＊（d） = Ω（n） involving SL ＊（n） and prime factor function ,and the specific forms of all the positive integer solutions were obtained .