Building on the base 3 findings

I started poking around with the maximum odd values in the sequence of numbers. An extremely robust pattern occurs. It turns out that the maximum odd values are always of the form 4*(sum of powers of 3) + 1.

For instance, the number 27 makes its way through the number 3077 before falling back to 1. 3077 = 4(3^0 + 3^1 + 3^2 + 3^3 + 3^4 + 3^6) + 1.

219 goes through 557 before falling.

557 = 4(3^0 + 3^1 + 3^4 + 3^4) + 1.

79 goes through 269 and then plummets.

269 = 4(3^0 + 3^1 + 3^3 + 3^3 + 3^4) + 1.

I must say, I'm excited to see this. If this is a canonical form which can work for both the multiplication and division steps and stay reasonable, then this might lead to something great.