Abstract
For any positive integer n, the Smarandache reciprocal function Sc(n) is deflned as Sc(n) = maxfm : y j n! for all 1 • ym; and m + 1 y n!g. That is, Sc(n) is the largest positive integer m such that y j n! for all integers 1 • ym. 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 inflnite group positive integers (m1; m2; ¢¢¢ ; mk) such that the equation Sc(m1 + m2 + ¢¢¢ + mk) = Sc(m1) + Sc(m2) + ¢¢¢ + Sc(mk):
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