An Inequality of the Smarandache Function (Report‪)‬

Abstract For any positive integer n, the famous Smarandache function S(n) is defined as the smallest positive integer m such that n|m!. That is, S(n) = min{m : m [element] N, n|m!}. In an unpublished paper, Dr. Kenichiro Kashihara asked us to solve the following inequalities S ([x.sup.n.sub.1]) + S ([x.sup.n.sub.2]) + ... + S ([x.sup.n.sub.n]) [greater than or equal to] nS ([x.sub.1]) * S ([x.sub.1]) ... S ([x.sub.n]).