Hallo Joseph,

the question looks similar to the Collatz Conjecture, but it is much easier to prove.

My proof is in the annex. But I guess my proof is not helpful for the real Collatz Conjecture.

With greetings from Germany

Gunter Lassmann

The proof is a pdf-file. If you want a tex-file please ask.

Thank you for this beautiful question

I've got almost the same proof.

I still think that formula

a_k=2(((3/2)^k)(n+1)-1) (*)

under the assumption

Odd(a_k)=a_k/2 for all k>1 and all odd n>1

could simplify Gunter's proof:

since a_k is integer and even, then

((3/2)^k)(n+1) is integer and

n+1 is divided by 2^k for all k>1 and all odd n>1.

It contradicts the fact that

for any odd n>1 there exists k>0 such that

(n+1)/ (2^k)n

such that Odd(a_k)