On the powerful and squarefree parts of an integer

Cloutier, Maurice-Étienne ; De Koninck, Jean-Marie ; Doyon, Nicolas

Journal of integer sequences, Tome 17 (2014) no. 8.

Résumé

Summary: 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 $m$ is a powerful number and $r$ a squarefree number, with $gcd(m,r)=1$. 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 $\eta1$. 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 $y=2\log x$.

Détail

Classification : 11A25, 11N37
Keywords: powerful numbers, arithmetic function