Sequence of nonzero digits with sum dividing decimal representation

Question

Is there an infinite sequence of nonzero digits $a_1,a_2,\ldots$ such that $$a_1+a_2+\ldots+a_n\mid\overline{a_1a_2\ldots a_n}$$ for all $n\geq 1$, where $\overline{a_1a_2\ldots a_n}$ denotes the number in decimal representation?

Answer 1

Observe that if there is no such sequence of length $m$, then there is no such sequence of length $m'>m$. This makes the problem amenable to brute force search by computer. It turns out the longest sequence is 24786. There is no sequence of length $6$, so that's it.