A Unified Approach to the Sandor-Smarandache Function

Abstract

The Sandor-Smarandache function, SS(n), is a recently introduced Smarandache-type arithmetic function, which involves binomial coefficients. It is known that SS(n) does not possess many of the common properties of the classical arithmetic functions of the theory of numbers. Sandor gave the expression of SS(n) when n ( ³ 3) is an odd integer. It is found that SS(n) has a simple form when n is even and not divisible by 3. In the previous papers, some closed-form expressions of SS(n) have been derived for some particular cases of n. This paper continues to find more forms of SS(n), starting from the function SS(24m). Particular attention is given to finding necessary and sufficient conditions such that SS(n) = n–5 and SS(n) = n–6. Based on the properties of SS(n), some interesting Diophantine equations have been studied. The study reveals that the form of SS(n) depends on the prime factors of the integer n in the natural order of the primes.