Re: Proof of maximum value?

It looks good to me. What you've proven is that it must be less than or equal to this bound. Is there an element that attains this bound?

Re: Proof of maximum value?

Yes, it turns out that the minimum element is always in the set. So, for all , . And it is simple to check that when .

Edit:

I forgot to mention that the minimum element always attains the maximum bound for that ratio since:

Re: Proof of maximum value?

Good! What is this problem related to, if I may ask?

Re: Proof of maximum value?

Let be the set of odd positive integers. Define by where is the largest power of 2 that divides . The function is sometimes called the Syracuse function, and is one of the methods used to investigate the 3n+1 Conjecture (also known as the Syracuse problem, the Collatz Conjecture, etc.).

Let . Notice that . Now, if we want to check the asymptotic density of in , we need to figure out how many elements of are less than for some . So, we have:

I now know that the second term on the left is bounded below by 0 and bounded above by .

Re: Proof of maximum value?

I knew it! I had a feeling it had something to do with the Collatz problem. All these 3's and 2's...

Re: Proof of maximum value?

I work on it whenever I get frustrated with my coursework. I find it relaxing to work on a problem that I don't care if I make any progress on. It helps relieve stress from grad school.