Revisiting an interesting number theory pattern

Hi /r/math,

I picked up a copy of Mathematica today, and as a learning exercise decided to do a little investigation on a pattern I found a couple years back. I’m here to share what I found and ask for advice on where to go next.

The problem considers a function F, which takes as input a natural number n and outputs the unique positive real number r such that 1r + 2r + 3r + … + (n-1)r = nr. So for instance F(2) = 0, F(3) = 1, F(4) is about 1.7305, and so on.

What I found a few years back was that the sequence of differences F(n+1) – F(n) appeared to tend to log(2). Today, I verified that the differences up to F(1000) do seem to approach log(2), and that furthermore F(n) seems to approach [; left(n – frac{3}{2} right) log{2} ;]. For instance, F(1000) = 692.10745709…, and 998.5 * log(2) = 692.10745978… Accuracy at such high values of n leads to me to think that F(n) really does approach log(2) (n-3/2).

What, if any, tools can I use to prove or disprove this? Is this related to any other areas / can someone refer me to any resources regarding this sort of pattern? Thanks in advance!

submitted by /u/no_elaboration 