一个新的Smarandache函数的均值

On the Mean Value of a New Smarandache Function

对任意正整数n,定义一个新的Smarandache函数D(n)=max{ab:a,b∈N,n=a(a+1)/2+b},其中N为所有正整数集合.利用初等方法和D(n)的性质,研究了函数D(n)的均值性质,并给出该函数各种均值的一个较强的渐近公式.

For any positive integer n, a new Smarandache function D（n） is defined as the largest positive integers ab such that n=a（a＋1）/2＋b.That is,D（n） =max {ab.a,b∈a（a＋1）/2 ＋b},where N denotes the set of all positive integers. Using the elementary method and the properties of the function D（n）, the mean value properties of D（n） are studied, several interesting mean value formulae are given,and an exact asymptotic formulae for the various mean value of the function D（n） is obtained.