To be clear

[sniffnoy]

Since I don't think I've stated it before explicitly here, here's the final form for what is really, I think, the main thing I've proved:

Let f(n) be the smallest number of ones needed to write n using addition and multiplication.

A number n is greater than all numbers of strictly lower f, if and only if it can be written as either 2^a3^k for some a,k with a≤10, or it can be written as 2^a(2^b3^m+1)3^k for some a,b,m,k with a+b≤2.