I was trying to find numbers with same number of odd and even divisors. The solution is $2l$ where $l$ is odd and I think I understand this - you're adding one even divisor for every odd one that was there.
If you take power of 2 and multiply by 2 you increase the number of even divisors by 1. This also makes sense to me since you keep all the previous divisors and add one new.
(paragraph below NOT true, you get new even divisor for every odd one, see comment)
But it looks like (I don't know if this is actually true) that if you multiply any even number that is not power of 2 by 2 you increase the number of even divisors by 2 (while the number of odd divisors doesn't change).
So what's happening here? Can this be more generalized (number of divisors with respect to parity)?
It looks to me you get one new divisor - the new number (makes sense) and another one for some other number that doesn't "overlap" (2 -> 4 -> 8 -> 16 -> ... when you start with number 6 and multiply by 2).
I hope this question makes sense but please ask for clarifications if not. It's also possible I'm just looking at numbers for too long.