Permutations of the set $\{1,2,...,n\}$ and prime numbers

I just observed for some small $n$ that we can find a permutation of the set $\{1,2,...,n\}$ which is such that sum of any two adjacent numbers is a prime number.

Take for example set $\{1,2,3,4,5,6\}$ and permute it to get $\{1,4,3,2,5,6\}$. Then we have $1+4=5$ and $4+3=7$ and $3+2=5$ and $2+5=7$ and $5+6=11$ so the sum of any two adjacent numbers is a prime number.

So naturally I asked myself could it be that there is some infinite set $\{n_i:i \in \mathbb N\}$, a subset of the set of natural numbers, such that every set of these sets $\{\{1,2,...,n_i\}:i \in \mathbb N\}$ can be permuted in at least one way to obtain a set that has the property that the sum of every two adjacent numbers is a prime number.

Can it be that such set exists? Or some known or conjectured fact forbids its existence?

This is still incomplete, but it may help:

We say $n$ is prime-permutable if we can permute $\{1,2,3\dots n\}$ so that the sum of adjacent terms is prime.

We wish to determine wheter the set $M$ of prime-permutable integers is finite or infinite. ( but if we can it would be great).

If it is finite we probably won't be able to prove it since if $n+1$ and $n+3$ are twin primes then $n$ is permutable, via the permutation:

$1,n,3,n-2,5,\dots, n-1,2$.

The number of suitable permutations for $n\leq 12$ is:

$ 2 : 2 $

$ 3 : 2 $

$ 4 : 8 $

$ 5 : 4 $

$ 6 : 16 $

$ 7 : 24 $

$ 8 : 60 $

$ 9 : 140 $

$ 10 : 1328 $

$ 11 : 2144 $

$ 12 : 17536 $

$ 13 : 23296 $

$ 14 : 74216 $

$ 15 : 191544 $

$ 16 : 2119632 $

$ 17 : 4094976 $

$ 18 : 24223424 $

$ 19 : 45604056 $