[bizaro86]

psyang said my series was missing one so I recalculated it. Once again no solution, but a few comments someone might find helpful (or completely misleading!) but I did compute the f(n) for the first 100 values of n

Spoiler!

The values where f(n)=n for the first 100 positive integers are:

{1,3,5,7,9,15,17,21,27,31,33,45,51,63,65,73,85,93, 99}

I'm more confident in this than I was in my previous hand calculated set as I did it in Excel, which doesn't make computational errors (and I did previously, which is why my last set was missing 21).

If anyone thinks the values of f(n) for the other values of n would be helpful I can post the entire table, but I don't see a use case for those.

The gaps in the pattern are not predictable in an obvious way. I tried to curve fit this data set to a variety of functions and got no useable results. I also tried checking just the results from the different piecewise portions of the function (equations 4/5) and had no useable results there.

The only observation I have that might be of any value is that the set of numbers above only has adjacent odd numbers when the even number between them is a power of 2.

ie
1/3 have 2^1=2 between them
3/5 have 2^2=4 between them
7/9 have 2^3=8 between them
15/17 have 2^4=16 between them
31/33 have 2^5=32 between them
63/65 have 2^6=64 between them

I will run this out to see if that holds.

It does. I went to 150 and you get 107,119,127,129.

so 127/129 have 2^7=128 between them. I doubt that's a coincidence, but it doesn't explain the other values in the set. More thought required for sure.