The Collatz Connection

There is one integer sequence, which is keeping me up at night sometimes if I start thinking about it. It is the sequence of odd integers, which do not have 5 as a Collatz exit point. Why don't they? What is so special about them? But most importantly, how can we predict the next term of this sequence? What is the formula? This sequence is not in OEIS. The first few terms that I found are:


Not found in OEIS
The Collatz Conjecture is important for integer factorization. I recently read a very nice explanation of why this problem is important. I came up with a couple of interesting sequences in this vicinity: A276260 and A276290 but the sequence described above is something that troubles me in a mathematical sense.

On a personal note, I just learned today that the mathematician who first thought of the Collatz problem died while attending a Mathematics Conference in the city I was born in.