The number $2$ has the interesting property that whenever $n>1$ is an integer, then $n \nmid (2^n-1)$. (It's a good exercise to prove this statement.)


Let's call a positive integer $b$ *$2$-like* if for all integers $n>1$ we have $n\nmid (b^n-1)$, and let's call it *almost $2$-like* if for all integers $n>1$ *except finitely many* we have $n\nmid (b^n-1)$.


**Question.** Is the collection of almost $2$-like numbers a proper superset of the collection of $2$-like numbers?



Integers $b$ such that $n \nmid (b^n-1)$ for $n>1$
$b=2$ is the only almost 2-like number. Indeed, if $n\mid (b^n-1)$ and $p$ is a prime divisor of $(b^n-1)/n$, then $np\mid (b^{np}-1)$. That is, existence of one $n>1$ dividing $b^n-1$ implies existence of infinitely many of them.
Also, for $b>2$, there exist at least one such $n$, e.g., $n=b-1$.


