Does there exist a sequence consists of infinite positive integers $a_1,a_2,…$ such that the sum of two arbitrary different numbers in the sequence is always coprime with the sum of three arbitrary different numbers in the sequence?

Assume that there exist such sequence.

For every prime $p$, if there exist three numbers from the sequence $a_i,a_j,a_k$ that are divisible by $p$, then $a_i+a_j$ is not coprime with $a_i+a_j+a_k$, contradiction. Thus $p$ can divide at most two numbers from the sequence. Therefore there are at most two even numbers from the sequence and there are infinite odd numbers from that sequence.

However, if there is at least one even numbers, then the sum of two odd numbers is not coprime with the sum of two odd numbers and an even one, since both of the sum are greater than $2$ and are both even. Hence there are no even numbers in the sequence.

Here I am stuck. How can I progress ? Is there a better way to solve the problem ?

(My other approach is to consider numbers of the form $m^n+1$, but it seems hopeless.)