Answering [this question][1], another question came to my mind. For which $n$ will the greedy algorithm work? 

We define a sequence of natural numbers $x_n$ recursively:
$$x_1 =1,$$
$$x_n \mbox{ is the smallest natural number such that } x_n+x_{n-1} \mbox{ is prime and } x_n\neq x_k \mbox{ for all } k<n.$$
We call $n$ ***wabbity*** if 
$$\{x_1,x_2,\ldots x_n\} = \{1,2,\ldots n\}.$$

Is there an infinitely many wabbity numbers? Is there an effective characterisation of wabbity numbers?


  [1]: https://mathoverflow.net/questions/241602/permutations-of-the-set-1-2-n-and-prime-numbers/241730#241730