Smallest Arguments for given Multiplicative Persistence

Sequence
The sequence of the smallest positive integers with successive multiplicative persistence begins as follows:
 * {| border="1"


 * 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" | $277 \, 777 \, 788 \, 888 \, 899$
 * }