Bounding the Smarandache function

Let S(n), for n ∈ N+ denote the Smarandache function, then S(n) is defined as the smallest m ∈ N+, with n|m!. From the definition one can easily deduce that if n = p1 1 p α2 2 . . . p αk k is the canonical prime factorization of n, then S(n) = max{S(p αi i )}, where the maximum is taken over the i’s from 1 to k. This observation illustrates the importance of being able to calculate the Smarandache function for prime powers. This paper will be considering that process. We will give an upper and lower bound for S(pα) in Theorem 1.4. A recursive procedure of calculating S(pα) is then given in Proposition 1.8. Before preceeding we offer these trivial observations: