  The $j$th divisor function $d_j$, which counts the ordered factorisations of
a positive integer into $j$ positive integer factors, is a very well-known
arithmetic function; in particular, $d_2(n)$ gives the number of divisors of
$n$. However, the $j$th non-trivial divisor function $c_j$, which counts the
ordered proper factorisations of a positive integer into $j$ factors, each of
which is greater than or equal to 2, is rather less well-studied. We also
consider associated divisor functions $c_j^{(r)}$, whose definition is
motivated by the sum-over divisors recurrence for $d_j$. After reviewing
properties of $d_j$, we study analogous properties of $c_j$ and $c_j^{(r)}$,
specifically regarding their Dirichlet series and generating functions, as well
as representations in terms of binomial coefficient sums and hypergeometric
series. We also express their ratios as binomial coefficient sums and
hypergeometric series, and find explicit Dirichlet series and Euler products in
some cases. As an illustrative application of the non-trivial and associated
divisor functions, we show how they can be used to count principal reversible
square matrices of the type considered by Ollerenshaw and Br\'ee, and hence
sum-and-distance systems of integers.
