Not sure if this question is formulated correctly. I am reading:
For all
This would mean that for a even, any two elements in the sequence should be relatively prime without exception: . As you can see, the conjecture is simply wrong.
Theorem: WLOG, for
Proof: Start by noticing -- see proof at http://www.mathhelpforum.com/math-he...s-2-n-1-a.html. So we can rewrite as , for and some x. It can be easily seen that k has the opposite parity of a. So if a is even, k is odd, and . If a is odd, k is even, and . QED