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 $$1, 2, 4, 6, 3, 10, 12, 7, 16, 18, 11, 22, 13, 5, 28$$ I conjecture that $a(n)+1$ is prime if and only if $a(n)=2(n-1)$.

Is there a way to prove it?