On an equation involving the Smarandache reciprocal function and its positive integer solutions

摘要

For any positive integer n, the Smarandache reciprocal function Sc(n) is defined as Sc(n) = max{m : y|n! for all 1≤y≤m, and m+1+n!}. That is, Sc(n) is the largest positive integer m such that y|n! for all integers 1≤y≤m. The main purpose of this paper is using the elementary method and the Vinogradov's important work to prove the following conclusion: For any positive integer k≥3, there exist infinite group positive integers (m1, m2,…, mk) such that the equation Sc(m1+m2+…+mk)=Sc(m1+Sc(m2)+…+Sc(mk).This solved a problem posed by Zhang Wenpeng during the Fourth International Conference on Number Theory and the Smarandache Problems.