What is known about these arithmetical functions? Let $n=\prod_p p^{c_p}$, $N\in \mathbb N$ and 
$$
\alpha_N(n)=\prod_p p^{c_p \bmod N}.
$$
The function $\alpha_N$ is multiplicative since $\alpha_N(n)\alpha_N(m)=\alpha_N(nm)$ for co-prime $n$ and $m$ and completely multiplicative for a subset of $\mathbb N$.
Further let
 $$\log \alpha_N(n)=
A_N(n)=\sum_p (c_p \bmod N) \log p.
$$
 It also reminds me to a sum of an adapted kind of von Mangoldt function, with following definition:
$$
    \Lambda^\star(n) = \begin{cases} \log p & \text{if }n=p^c \text{ for some prime } p \text{ and integer } c \ge 1 \text{ and } c\bmod N =1, \\ 0 & \text{otherwise.} \end{cases} 
$$
PS: Since the taste of the question was changed, some of the comments might be misleading. Sorry for that...
 A: I would call this function the $N^{th}$-power free part of $n$, which generalizes the $N=2$ case of the squarefree part of an integer.
Specifically, we have that $$\alpha_N(n)=n\prod_{p^{kN}|n}\left(\frac{1}{p^{N}}\right),$$
and $\alpha_N(n)$ removes all of the $N^{th}$ powers that divide $n$ leaving behind the $N^{th}$-power free part. It will share many properties with the squarefree part. For example, $\alpha_N(n)=n$ for any $N^{th}$-power free integer, which is a set of density $\frac{1}{\zeta(N)}$ in the integers. The average of $\alpha_N(n)$ can be computing using the techniques in the answer Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function. Since $$\mu*\frac{\alpha_{N}(n)}{n}=\begin{cases}
1 & k=0\\
0 & k\neq0\ \text{mod}\ N\\
-\frac{p^{N}-1}{p^{aN}} & k=aN
\end{cases}
 $$ and we find that $$\sum_{n\leq x}\alpha_N(n)=\frac{x^2}{2}\frac{\zeta(2N)}{\zeta(N)}+O(x\log x).$$ (The error term is $O(x)$ for all $N\geq 3$.) In particular, when $N=2$, we find that the squarefree part of $n$ has average $$\sum_{n\leq x}\alpha_2(n)=\sum_{n\leq x} \text{squarefree}(n)=\frac{x^2\pi^2}{30}+O(x\log x).$$ Similarly, the fourth-powerfree part of $n$ has average $$\sum_{n\leq x}\alpha_4(n)=\sum_{n\leq x}\text{fourth-powerfree-part}(n)=\frac{\pi^4x^2}{210}+O(x).$$
