polydivisible number

Given a base $b$ integer $n$ with $k$ digits $d_1,\mathrm{\dots },d_k$, consider $d_k$ the least significant digit and $d_1$, to suit our purpose in this case. If for each it is the case that

$$(\sum _{i=1}^j{d}_i{b}^{k-j-i})|j,$$

then $n$ is said to be a polydivisible number.

A reasonably good estimate of how many polydivisible numbers base $b$ has is

$$\sum _{i=2}^{b-1}\frac{(b-1)b^{i-1}}{i!}.$$

In any given base, there is only one polydivisible number that is also a pandigital number.