Research New Pseudo Smarandache Function And Its Mean Value Category

Prof.F.Smarandache is a Romanian-American specialist in the field of num-bcr theroy.He proposed the Smarandache function in his book Only problem-s,not solutions.For any positive integer n,the Smarandache function S(n) is defined as the smallest positive integer m that makes n|m!.That is S(n)=min{m:m∈N,n|m!}David Gorski proposed the pseudo Smaran-dachc function according to the Smarandache function.∨n∈N,the pseudo Smarandache function Z(n)is defined as the smallest positive integer k that makes n|1＋2＋3＋…＋k.That is Z(n)=min{k:k:∈N,n|k(k＋1)/2).A few clementary propertics of thc pseudo Smarandache function were gottcn us-ing elementary methods by David Gorski.To study the Smarandache function furthcr,some extendcd pseudo Smarandachc functions were proposed and a few conclusions were achieved.Based on the background above,this paper mainly studics the following issues.1. A new pseudo Smarandachc function zo(n)is defined.When n is an even number,zo(n)=m.m is the smallest positive integer that makes n|2＋4＋6＋…＋2m=m(m+1).That is zo(n)=min{m:m∈N,n|m(m+1)}. When n is an odd number,m is the smallest positive integer that makes n|1+3+5+...+(2m一1)=m2.That is zo(n)=min{m:m∈N, n|m2}.Analytic methods and Perron formula are used to study its mean value properties,and a sharp asymptotic formula for it is given.2. The pseudo Smarandachc function is extended and a new pseudo S-marandache function T(n)=min{a＋b:a,b∈N,n|a(s＋1)+b} is introduced. Basic properties of this function are researched by using elementary methods. Asymptotic formulas for several kinds of mean values of this function are given.