What's this concatenated number's starter?

A number of programming languages construct large integers through 'concatenating' the digit to the end of the existing number. For example, Labyrinth, or Adapt. By concatenating the digit to the end, I mean that, if the existing number is \$45\$, and the digit is \$7\$, the result number is \$457\:(45 \times 10 + 7)\$.

A concatenated number is a number than be built this way by the use of the multiples of single digit numbers: \$1, 2, 3, 4, 5, 6, 7, 8, 9\$ A.K.A an element in one of these 9 sequences:

$$1, 12, 123, 1234, \: \dots$$ $$2, 24, 246, 2468, 24690, \: \dots (\text{note } 24690 = 2468 \times10+10\text{, not } 246810)$$ $$3, 36, 369, 3702, 37035$$

(work in progress, will finish later)