Consider a number $a_1a_2a_3a_4 \dots a_n$ in some base $b$, such that for each $k, 1\leq k \leq n$, the subnumber $a_1a_2\dots a_k$ is a multiple of $k$.

For instance $1836$ is such a number in base $10$, because $1$ is a multiple of $1$, $18$ is a multiple of $2$, $183$ is a multiple of $3$, and $1836$ is a multiple of $4$.

Let $N(b)$ be the maximum possible value of $n$ for base $b$.

How large is $N(b)$?

We might expect $N(b)$ to be about $eb$. Indeed, there are about $b^n/n!$ such numbers of length $n$, which by Stirling's approximation goes below $1$ sometime around $n=eb$.

The only lower bound I have is that $N(b)\geq b$. I don't have any upper bound at all.

This question is the result of a conversation with John Conway.