For any positive integer n, the famous Pseudo Smarandache Square-free function Zw(n) is defined as the smallest positive integer m such that m n is divisible by n. That is, Zw(n) = min{m : n|m n , m ∈ N }, where N denotes the set of all positive integers. The main purpose of this paper is using the elementary method to study the properties of Zw(n), and give an inequality for it. At the same time, we also study the solvability of an equation involving the Pseudo Smarandache Square-free

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... and prove that it has infinity positive integer solutions. Keywords The Pseudo Smarandache Square-free function, Vinogradov's three-primes theorem , inequality, equation, positive integer solution. §1. Introduction and results For any positive integer n, the famous Pseudo Smarandache Square-free function Z w (n) is defined as the smallest positive integer m such that m n is divisible by n. That is, Z w (n) = min{m : n|m n , m ∈ N }, where N denotes the set of all positive integers. This function was proposed by Professor F. Smarandache in reference [1], where he asked us to study the properties of Z w (n). From the definition of Z w (n) we can easily get the following conclusions: If n = p α1 1 p α2 2 · · · p αr r denotes the factorization of n into prime powers, then Z w (n) = p 1 p 2 · · · p r. From this we can get the first few values of Z w (n) are: Z w (1) = 1, Z w (2) = 2, Z w (3) = 3, Z w (4) = 2, Z w (5) = 5, Z w (6) = 6, Z w (7) = 7, Z w (8) = 2, Z w (9) = 3, Z w (10) = 10, · · ·. About the elementary properties of Z w (n), some authors had studied it, and obtained some interesting results, see references [2], [3] and [4]. For example, Maohua Le [3] proved that ∞ n=1