optimal number of divisors for powers of 2

Question

I am running an experiment where i need to calculate the value close to a power of 2 that results in the largest number of divisors. Specifically i am testing values in the range of 1% less than the value of the power of 2. I have found the following relationship for values $2^n$

n   l
9   510
10  1020
11  2040
12  4080
13  8190
14  16380
15  32760
16  65520
17  131040

where $l$ is the value close to the value of the power of 2 that yields the greatest number of divisors. There seems to be a pattern. Is anyone aware of some theory that might explain this?

Answer 1

Having a large number of divisors and being close to a power of $2$ are conflicting objectives so that an "optimal number" seems elusive. You chose an arbitrary limit of $1\%$ which I doubt makes things simpler. IMO, the irregularity of the function $\sigma(n)$, which resists to analysis, makes your question too difficult.