Abstract For any positive integer n, the famous Pseudo Smarandache Square-free function [Z.sub.w](n) is defined as the smallest positive integer m such that mn is divisible by n. That is, [Z.sub.w](n) = min{m : n|[m.sup.n],m [member of] 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 [Z.sub.w](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 function, 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.

[subsection]1. Introduction and results

For any positive integer n, the famous Pseudo Smarandache Square-free function [Z.sub.w](n) is defined as the smallest positive integer m such that mn is divisible by n. That is,

[Z.sub.w](n) = minfm : n|[m.sup.n], m [member of] 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.sub.w](n). From the definition of [Z.sub.w](n) we can easily get the following conclusions: If n = [p.sup.[alpha]1] [p.sup.[alpha2]2] ... [p.sup.[alpha]r]r denotes the factorization of n into prime powers, then [Z.sub.w](n) = p1p2 ... pr. From this we can get the first few values of [Z.sub.w](n) are: [Z.sub.w](1) = 1, [Z.sub.w](2) = 2, [Z.sub.w](3) = 3, [Z.sub.w](4) = 2, [Z.sub.w](5) = 5, [Z.sub.w](6) = 6, [Z.sub.w](7) = 7, [Z.sub.w](8) = 2, [Z.sub.w](9) = 3, [Z.sub.w](10) = 10, ... . About the elementary properties of [Z.sub.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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is divergence. Huaning Liu [4] proved that for any real numbers [alpha] [greater than or equal to] 0 and x [less than or equal to] 1, we have the asymptotic formula

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where [zeta](s) is the Riemann zeta-function. between [Z.sub.w] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, for any positive integer k [greater than or equal to] 1, we consider the relationship between [Z.sub.w] x k Yi=1

and [Z.sub.w](mi). In reference [2], Felice Russo suggested us to study the relationship between them. For this...