Smallest Arguments for given Multiplicative Persistence

Sequence
Let $P \left({n}\right)$ denote the multiplicative persistence of a natural number $n$.

Let $a: \N \to \N$ be the partial mapping defined as:
 * $\forall n \in \N: a \left({n}\right) = \text{the smallest $m \in \N$ such that $P \left({m}\right) = n$}$

The sequence of $a \left({n}\right)$ for successive $n$ begins as follows:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $a \left({n}\right)$
 * align="right" style = "padding: 2px 10px" | $0$
 * align="right" style = "padding: 2px 10px" | $0$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $25$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $39$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $77$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $679$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $6788$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $68 \, 889$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $2 \, 677 \, 889$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $26 \, 888 \, 999$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $3 \, 778 \, 888 \, 999$
 * align="right" style = "padding: 2px 10px" | $11$
 * align="right" style = "padding: 2px 10px" | $277 \, 777 \, 788 \, 888 \, 899$
 * }
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $2 \, 677 \, 889$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $26 \, 888 \, 999$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $3 \, 778 \, 888 \, 999$
 * align="right" style = "padding: 2px 10px" | $11$
 * align="right" style = "padding: 2px 10px" | $277 \, 777 \, 788 \, 888 \, 899$
 * }
 * align="right" style = "padding: 2px 10px" | $11$
 * align="right" style = "padding: 2px 10px" | $277 \, 777 \, 788 \, 888 \, 899$
 * }

It is not known what $a \left({12}\right)$ is, but it is known to be greater than $10^{200}$.