suffices to do so when $n$ is equal to a prime $p$.

In this paper we investigate the *average* behaviour of $f(p)$ for $p$ a prime. More precisely, we consider the asymptotic behaviour of the sum $$\sum_{p{\leqslant}N} f(p)$$ where $N$ is a large parameter, and $p$ ranges over all primes up to $N$. As we are only interested in asymptotics, we may ignore the case $p=2$, and focus on the odd primes $p$.

Let us call a solution $(x,y,z)$ to a *Type I solution* if $n$ divides $x$ but is coprime to $y,z$, and a *Type II solution* if $n$ divides $y,z$ but is coprime to $x$. Let $f_{{\operatorname{I}}}(n), f_{{{\operatorname{II}}}}(n)$ denote the number of Type I and Type II solutions respectively. By permuting the $x,y,z$ we clearly have $$\label{3ff}
 f(n) {\geqslant}3f_{{\operatorname{I}}}(n) + 3f_{{\operatorname{II}}}(n)$$ for all $n>1$. Conversely, when $p$ is an odd prime, it is clear from considering the denominators in the Diophantine equation $$\label{4pn}
\frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$ that at least one of $x,y,z$ must be divisible by $p$; also, it is not possible for all three of $x,y,z$ to be divisible by $p$ as this forces the right-hand side of to be at most $3/p$. 