Product of powers of two less than or equal to 2n

Question

Question 1

Is there an equation to calculate or approximate the Product of powers of two less than or equal to in general 2n?

Example 1

Given for example 2n = 14 this would simply be $2.4.8=2.2^2.2^3$ easily seen by inspection.

I know one can use Legendre's Theorem (see question https://math.stackexchange.com/a/141197/155629) to do this but that's not what I'm after since it uses the integer function [x]; same with answer below. An approximation would be fine too.

Answer 1

The product will be of the form $2.2^2.2^3\dots2^{m-1}.2^m$ for some $m$, which simplifies to $2^{m(m+1)/2}$. And $m$ is the exponent of the largest power of $2$ that is less than or equal to $2n$ so

$m = \lfloor \log_2 (2n) \rfloor = 1 + \lfloor \log_2 (n) \rfloor$

In the example,

$2n=14$

$n=7$

$m=1+\lfloor 2.807\dots \rfloor=3$

$m(m+1)/2 = 6$

and so the product is $2^6=64$.