I would like to issue a challenge for the following problem:
Give a closed formula for term $a_n$ of an increasing sequence of positive integers $a_1,a_2,\dots,a_n$ such that $(a_i,a_j)=1$ for all $i,j$.
We say that one sequence $a_n$ is better than another $(b_n)$ if there exist $N$ such that $a_n\leq b_n$ for all $n\geq N$.
Right now the only example I have is $2^{2^n}+1$ from this question, but I look forward to contributing when I find something "better"
This problem turns out to not have a particularly well defined solution. One can do arbitrarily well at this game using any sequence that satisfies the assumptions. Consider the example of the the Fibonacci numbers, which are given by the well-known formula $F_n=\phi^n+(1-\phi)^n$. Given any sequence $a_n$ that only produces relatively prime numbers and has a closed form $\sigma(n)$ we can now construct a sequence that beats $a_n$ by looking at $F_{a_n}:=\phi^{\sigma(n)}+(1-\phi)^{\sigma(n)}$. Since $F_i \geq i$ holds for all positive $i$, $F_{a_n}\geq a_n$ holds for all $n$.
There might be a complexity theory-esque way to look at rates of growth of self-convolution though