  Under sufficiently strong assumptions about the first term in an arithmetic
progression, we prove that for any integer $a$, there are infinitely many $n\in
\mathbb N$ such that for each prime factor $p|n$, we have $p-a|n-a$. This can
be seen as a generalization of Carmichael numbers, which are integers $n$ such
that $p-1|n-1$ for every $p|n$.
