Recently, in high-school competition maths, I came across such a question:
Does there exist an integer $n$, with $2000$ factors, such that $n$ divides $2^n+1$, or:
$$n\mid{2^n+1}?$$ This question is tagged as a practice question for number theory and is part of a lecture on indefinite equations. The solution was not provided, and I could not seem to figure it out.
Any help would be much appreciated.
A useful lemma is that for all $k$ we have $$3^k\,|\,2^{3^k}+1$$
The proof is a straight forward induction, and may be found, e.g., here. Indeed, one can show a slightly stronger result than we require.
These are not all the $n$ such that $n\,|\,2^n+1$ but they suffice to answer this question, since $3^k$ has $k+1$ divisors.
Worth remarking: the $n$ for which $n\,|\,2^n+1$ which are not powers of $3$ form a rather erratic list. in OEIS they form sequence A016057.