This is similar to an existing thread, but sufficiently different, I think.

For any positive integer n, let $\displaystyle S(n)=\Sigma^n_{k=1}\left({1\over2k-1}-{1\over2k}\right)={u_n\over d_n}$ with $\displaystyle u_n$ and $\displaystyle d_n$ relatively prime. The question concerns $\displaystyle u_n$ and $\displaystyle d_n$.

One result. The exact power of 2 that divides $\displaystyle d_n$ is floor(log_{2}(n))+1.

Here's the proof:

What else can one say about the sequence $\displaystyle d_n$? The exact power of 3 that divides $\displaystyle d_n$ seems to be an increasing sequence, but I can't find any closed formula as for 2. The exact power of 5 that divides $\displaystyle d_n$ is not even an increasing sequence.

The integers $\displaystyle u_n$ and $\displaystyle d_n$ get quite large. I know almost nothing about the $\displaystyle u_n$.

$\displaystyle u_2$, $\displaystyle u_3$, $\displaystyle u_5$, $\displaystyle u_8$ and $\displaystyle u_9$ are prime.

Java has a pretty sophisticated probalistic prime testing algorithm. Java says with probability greater than 1-one millionth that $\displaystyle u_{254}$ is prime; $\displaystyle u_{254}$ has 221 decimal digits.

(I inadvertenly attached a thumbnail and don't know how to delete it; ignore it.)