On factorizations into coprime parts

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the sums of $F(n)^β$ and $G(n)^β$ up to $x$ for all real $β$ and the asymptotic bounds for $f(n)^β$ and $g(n)^β$ for all negative $β$.

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