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A Proof of Sondow’s Conjecture on the Smarandache Function

A Proof of Sondow’s Conjecture on the Smarandache Function

The Smarandache function of a positive integer n, denoted by S(n), is defined to be the smallest positive integer j such that n divides the factorial j!. In this note, we prove that for any fixed number k > 1, the inequality nk<S(n)! holds for almost all positive integers n. …