parasitic number

Given a base $b$ integer $n$ with $k$ digits $d_1,\mathrm{\dots },d_k$ (with $d_1$ being the least significant digit), and an integer , if it is the case that

$$m(\sum _{i=1}^k{d}_i{b}^{i-1})=d_1{b}^{k-1}+(\sum _{i=2}^k{d}_i{b}^{i-2}),$$

then $n$ is called an $m$-parasitic number or $m$-left-transposable number.

If one takes an $m$-parasitic number, concatenates copies of its digits in order whatever number of times one wants, the resulting new number will also be $m$-parasitic.