I'm probably missing something here, but does that not only tell you that is coprime to all the other terms in the sequence and is the only term in the sequence with this property
No, it was me missing something (misreading the question, as so often). I was thinking that the sequence was meant to be a permutation of the natural numbers. But the question does not say that, so it is harder than I first thought. I doubt whether I'll have time to reconsider it on Christmas Day however.