Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $PS(2)$, then apply $P(3)$ to $PS(2)$ to get $PS(3)$, then apply $P(4)$ to $PS(3)$, etc. The limit of $PS(n)$ is $a(n)$ (A057063).

The sequence begins $$1, 2, 4, 6, 3, 10, 12, 7, 16, 18, 11, 22, 13, 5, 28$$

Some examples: $$1,2,(4,3),(6,5),(8,7),(10,9),(12,11),(14,13),(16,15),(18,17)$$ $$1,2,4,(6,5,3),(7,10,8),(12,11,9),(13,16,14),(18,17,15)$$ $$1,2,4,6,(3,7,10,5),(12,11,9,8),(16,14,18,13)$$ $$1,2,4,6,3,(10,5,12,11,7),(8,16,14,18,9)$$

I conjecture that $a(n)+1$ is prime if and only if $a(n)=2(n-1)$.

Is there a way to prove it?

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