Carmichael numbers of the form $2^n p^m+1$ with odd exponent $m$: refinements of a theorem of Florian Luca, a general independence lemma, and a computational study

Florian Luca devoted the last paper of his life to the set $\mathcal K$ of odd positive integers $k$ such that $2^n k+1$ is a Carmichael number for some positive integer $n$. He proved that if $m\ge 5$ is fixed and odd, then there are only finitely many primes $p$ with $p^m\in\mathcal K$. In this no