The number 1 has the property that GCD(1,j) = 1 for all j, and it is the only number with that property. Therefore .
The number 2 has the property that GCD(2,j) = 1 or 2 for all j, and it is the only number with that property. Therefore .
Can you continue that line of reasoning?
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.