Frequency of Greatest Integer quotients

[jleach]

When looking for potential factors of an integer, I noticed that I could predict the frequency of the greatest integer function quotient and use this prediction to jump to the next potential factor. See the attachment for an example. I don't know if someone else had already discovered this, but I thought that it was cool.

 

[hamster143]

The problem with the algorithm is that N/H-N/(H+1) will always evaluate to 1 for k<sqrt(N), so it will make you inspect all potential factors up to sqrt(N). Which are all the potential factors you need to inspect anyway.

 

[jleach]

You have a point. I think that it's still an interesting observation, but let me think on it and I may be able to find a modification that will improve the method.

 

[jleach]

Thanks for making me take another look, because I can already see something else that is interesting. If I look at the spacing between actual factors found by my method, there seems to be a symmetry. If this is real, then it could be useful.

 

[jleach]

Nothing comes to me yet. I'll just put it in my back pocket and maybe I'll find a use for it someday. Sorry, I was more interested in just sharing my observation than looking at how it could be applied, so I didn't catch that it didn't save any work in finding factors.

Finding the factors between two numbers is a real problem that I sometimes have to deal with, so if anyone knows a good algorithm, let me know.

As for as the frequency of greatest integer quotients and the way that it pulls in all of the factors so that they are symmetrically spaced, I still think that it's interesting.