Bounding the Smarandache Function

Let S (n), for n E N+ denote the Smarandache function, then S (n) is defined as the smallest m E N+, with nlm!. From the definition one can easily deduce that if n = prlp~2 .. . p~k is the canonical prime factorization of n, then Sen) = max{S(pfi)}, 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(pQ) in Theorem 1.4. A recursive procedure of calculating S(pQ) is then given in Proposition 1.8. Before preceeding we offer these trivial observations: Observation 1. Jfp is prime, then S(P) = p. Observation 2. Ifp is prime, then S(Pk) ~ kp. Observation 3. p divides S(pk) Observation 4. If p is prime and k < p, then S(;f) = kp. To see that observation 4 holds, one need only consider the sequence 2,3,4 ... ,p -1,p,p+ 1, ... , 2p,2p+ 1, ... ,3p, ... , kp and count the elements which have a factor of p. Define Tp(n) = L~l[~J, where [.J represents the greatest integer function. The function Tp counts the number of powers of pin n!. To relate Tp(n) and Sen) note that SCpO<) is the smallest n such that Tp(n) 2:: a. In other words SCpO<) is characterized by (*) Tp(S(pO<» 2:: a and Tp(S(PIl:) 1) ~ a 1. 1