Prime Cycle

Form a cycle with a permutation of the first $n$ positive integers . The cycle is called Prime Cycle if all neighboring pairs sum up to be a prime . The two distinct prime cycles for $n=6$ are:

• $1,4,3,2,5,6$
• $1,6,5,2,3,4$

The permutation $3,2,5,6,1,4$ is considered the same as the first sequence.

How many distinct prime cycles are there for $n=16$ ?