Duro Kurepa conjectured that the function on the title is always nonzero in $\mathbb{Z}/{n \mathbb{Z}}$
provided $n>2.$ Daniel Barsky
and B\'enali Benzahgou [MR2145571 (2006a:11025)]
proved this. Thus, for all odd prime numbers $p$
$$
ku(p) = 0!+1!+ \cdots + (p-1)! \pmod{p}
$$
is an inversible element of $\mathbb{Z}/{p\mathbb{Z}}.$


Question: There are infinite subsets $S$ of the set of odd composite numbers
such that for each $s \in S$ one has that $ku(s) \pmod{s}$ is inversible in $\mathbb{Z}/{s \mathbb{Z}}.$ ?



About Duro Kurepa function: $ku(n)=0!+1!+ \cdots+(n-1)! \pmod{n}$
Yes, if $s$ is a power of an odd prime then your function gives a unit mod $s$. 


