  Given an integer $k$, define $C_k$ as the set of integers $n > \max(k,0)$
such that $a^{n-k+1} \equiv a \pmod{n}$ holds for all integers $a$. We
establish various multiplicative properties of the elements in $C_k$ and give a
sufficient condition for the infinitude of $C_k$. Moreover, we prove that there
are finitely many elements in $C_k$ with one and two prime factors if and only
if $k>0$ and $k$ is prime. In addition, if all but two prime factors of $n \in
C_k$ are fixed, then there are finitely many elements in $C_k$, excluding
certain infinite families of $n$. We also give conjectures about the growth
rate of $C_k$ with numerical evidence. We explore a similar question when both
$a$ and $k$ are fixed and prove that for fixed integers $a \geq 2$ and $k$,
there are infinitely many integers $n$ such that $a^{n-k} \equiv 1 \pmod{n}$ if
and only if $(k,a) \neq (0,2)$ by building off the work of Kiss and Phong.
Finally, we discuss the multiplicative properties of positive integers $n$ such
that Carmichael function $\lambda(n)$ divides $n-k$.
