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Edit: Sorry, for the inconvenience: I have edited the question, since there was a misconception in my thinking. Let $P_1(n) := 1$ if $n=1$ and $\max_{q\mid n, \text{ } q\text{ prime}} q$ otherwise, ...

I was somewhat recently introduced to the Matula-Goebel bijection between rooted trees and natural numbers. (nicely illustrated here http://keithbriggs.info/matula.html) Looking through them, I ...