Finding infinite sequences with pairwise relatively prime outputs.

Question

I am looking for a formula which for every element in $\mathbb{Z}$ as an input, gives pairwise relatively prime outputs. That is for example thanks to Greg Martin's suggestion the positive outputs of $2^{2^{n}}+1$ seem to all be pairwise relatively prime. I understand that the infinite set of primes is a solution however there is no known formula for primes. I have also tried the product ($\prod_{i=1}^ni)+1$ Which I am 90% sure is correct. In the best of scenarios I can find a working solution of the form $y=f(x)$ and maybe a proof solidifying my guess. So far I have been unsuccessful but would like to hear your suggestions or answers. Thank you.

Answer 1

The Fermat numbers $2^{2^n}+1$ are pairwise relatively prime - this is not just an observation but is in fact provable.

It's easy to turn this into a function $f\colon \Bbb Z\to\Bbb N$ where the outputs are pairwise relatively prime: $$ f(n) = \begin{cases} 2^{2^{2n}}+1, &\text{if }n\ge0; \\ 2^{2^{2|n|-1}}+1, &\text{if }n\le-1. \end{cases} $$ Or, if you prefer something defined without cases, just let $f(n) = 2^{2^{|3n-1|}}+1$. Or if you even want an analytic function: $f(n) = 2^{2^{(3n-1)^2}}+1$.