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On the Powerful and Squarefree Parts of an
Integer Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon
Département de mathématiques et de statistique
Université Laval
Québec G1V 0A6
Canada
mailto:maurice-etienne.cloutier.1@ulaval.camaurice-etienne.cloutier.1@ulaval.ca
mailto:jmdk@mat.ulaval.cajmdk@mat.ulaval.ca
mailto:nicolas.doyon@mat.ulaval.canicolas.doyon@mat.ulaval.ca

in

Abstract:

Any integer $n\ge 2$ can be written in a unique way as the product of its powerful part and its squarefree part, that is, as n=mr where mis a powerful number and r a squarefree number, with (m,r)=1. We denote these two parts of an integer n by $\pow(n)$ and $\sq(n)$respectively, setting for convenience $\pow(1)=\sq(1)=1$. We first examine the behavior of the counting functions $\sum_{n\le x,\, {\scriptsize\sq}(n)\le y} 1$ and $\sum_{n\le x,\, {\scriptsize \pow}(n)\le y} 1$. Letting P(n) stand for the largest prime factor of n, we then provide asymptotic values of $A_y(x):=\sum_{n\le x,\, P(n)\le y} \pow(n)$ and $B_y(x) :=\sum_{n\le x,\, P(n)\le y} \sq(n)$when y=x^1/u with $u\ge 1$ fixed. We also examine the size of A_y(x) and B_y(x) when $y=(\log x)^\eta$ for some $\eta>1$. Finally, we prove that A_y(x) will coincide with B_y(x) in the sense that $\log(A_y(x)/x) = (1+o(1))\log(B_y(x)/x)$ as $x\to \infty$if we choose $y=2\log x$.