Moderator Message: this question is from an ongoing competition.
Define a prime $p$ as having $k$ hands in $n$'s hair if $p^k|n$ and $n|2^n+1$ . Does there exist an integer $n$ with $2012$ hands in its hair? Does there exist an integer $n$ with $2012$ distinct primes' hands in its hair?
Furthermore it would be fantastic if you could show your full logic, thank you.
For example, $3$ has 1 hand in its own hair, and I can't think of any other nice examples.