This question is related to my other question regarding entropy with respect to a given multiplicative function ( Limit for entropy of prime powers defined by multiplicative arithmetic function ). Call a sequence $(a_n)_{n \in \mathbb{N}}$ irreversible with respect to a multiplicative function $f\ge 1$ if $H_f(a_n) < H_f(a_{n+1})$ where $H_f$ is the entropy with respect to $f$. Let $P_n$ be the $n$-th primorial. Then it is not difficult to show, that the sequence $(P_n)_{n \in \mathbb{N}}$ is irreversible with respect to every multiplicative function $f \ge 1$. Hence this suggests, that given $P_{n+1}$ for a large $n$, then it will be difficult to compute $P_n$. So this motivates the question:
Is there a known fast algorithm to compute the primorial $P_{n}$ given the primorial $P_{n+1}$ (for a large value of $n$)? (We observe only the value of $P_{n+1}$ not the value of $n$)