Some identities involving the k-th power complements

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Authors: Yanni Liu and Jinping Ma
Date: June 2006
From: Scientia Magna(Vol. 2, Issue 2)
Publisher: Neutrosophic Sets and Systems
Document Type: Article
Length: 810 words
Lexile Measure: 1720L

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Abstract The main purpose of this paper is using the elementary method to study the calculating problem of one kind infinite series involving the k-th power complements, and obtain several interesting identities.

Keywords k-th power complements, Identities, Riemann-zeta function.

[section] 1. Introduction and Results

For any given natural number k [greater than or equal to] 2 and any positive integer n, we call [a.sub.k](n) as a k-th power complements, if [a.sub.k](n) is the smallest positive integer such that n x [a.sub.k](n) is a perfect k-th power. That is,

[a.sub.k](n) = min{m : mn = [u.sup.k], u [member of] N}.

Especially, we call [a.sub.2](n), [a.sub.3](n), [a.sub.4](n) as the square complement number, cubic complement number, and the quartic complement number, respectively. In reference [1], Professor F.Smarandache asked us to study the properties of the k-th power complement number sequence. About this problem, there are many people have studied it, see references [4], [5], and [6]. For example, Lou Yuanbing [7] gave an asymptotic formula involving the square complement number [a.sub.2](n) . Let real number x[greater than or equal to]3, he proved that

[summation over n [less than or equal to] d([a.sub.2](n)) = [c.sub.1]xlnx + [c.sub.2]x + O ([x.sup.1/2+[epsilon]]

where d(n) is the divisor function,...

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Gale Document Number: GALE|A155737596