[psyang]

Originally Posted by bizaro86

Another step that I think has to be relevant and/or a solution, but one that absolutely does not come with a proof.

Spoiler!

So, I calculated the set of f(n)=n above, and I've been trying to find patterns. Fact that +/- 1 of every power of 2 was in the list seemed relevant. I spent some time expressing each one in terms of the nearest powers of 2, when I realized that's basically binary.

The set is 1,3,5,7,9,15,17,21,27,31,33,45,51,63,65...

In binary that is

1, 11, 101, 111, 1001, 1111, 10001, 10101, 11111, 100001, 101101, 110011,111111, 1000001

The pattern is that they are all symmetrical.

So while I can't in any way prove it, f(n)=n for the positive integers that are symmetrical when expressed in binary.

Frankly, I'm amazed that such a function exists. If you asked me to do this the other way (find a function where this was true) I absolutely could not do it. (And I did spend some time thinking about whether that was a function, specifically did it have only one value of f(x) for every value of x)

This does fit my previous "no odds" observation, because every symmetrical number in binary ends with a 1, which precludes it from being even.

Anyway, this is past my abilities for an analytical solution. I tried to treat it like a system of equations and got nowhere. Empirical solutions are my specialty - but even I recognize that this is a problem that almost certainly has an elegant analytical solution that I am missing, because there's no way something like that happens as a weird happenstance- someone smarter than me created this problem by back calculating the function somehow.

Thanks for posting this one! I won't be able to make any more progress I dont think but I enjoyed the hunt, and it's not something I would have even believed was possible to create.

We have a winner! So glad you kept at it!

Spoiler!

I could tell you were on the right path, but looking at the binary representation of n isn't an obvious step. Finding the values before/after the powers of 2 was a great insight on your part.

I found a different pattern that lead to the discovery of what f actually does. I'll share when the solution to the followup below is found.

So, while you have correctly discovered the conditions on n for which f(n)=n, I'm curious if anyone would like to take a stab at what f actually does to n, and then to prove that that is the case.