On the pseudo Smarandache square-free function

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Vol. 4 (2008), No. 2, 21-24

On the pseudo Smarandache square-free function Bin Cheng Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China Abstract For any positive integer n, the famous Pseudo Smarandache Square-free function Zw (n) is defined as the smallest positive integer m such that mn is divisible by n. That is, Zw (n) = min{m : n|mn , 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 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.

§1. Introduction and results For any positive integer n, the famous Pseudo Smarandache Square-free function Zw (n) is defined as the smallest positive integer m such that mn is divisible by n. That is, Zw (n) = min{m : n|mn , 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 Zw (n). From the αr 1 α2 definition of Zw (n) we can easily get the following conclusions: If n = pα 1 p2 · · · pr denotes the factorization of n into prime powers, then Zw (n) = p1 p2 · · · pr . From this we can get the first few values of Zw (n) are: Zw (1) = 1, Zw (2) = 2, Zw (3) = 3, Zw (4) = 2, Zw (5) = 5, Zw (6) = 6, Zw (7) = 7, Zw (8) = 2, Zw (9) = 3, Zw (10) = 10, · · · . About the elementary properties of Zw (n), some authors had studied it, and obtained some interesting results, see references [2], [3] and [4]. For example, Maohua Le [3] proved that ∞ X

1 , α²R, α > 0 (Zw (n))α n=1 is divergence. Huaning Liu [4] proved that for any real numbers α > 0 and x ≥ 1, we have the asymptotic formula · ¸ ´ ³ X 1 ζ(α + 1)xα+1 Y 1 + O xα+ 2 +² , (Zw (n))α = 1− α ζ(2)(α + 1) p p (p + 1) n≤x


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