TITLE: Are there numbers that describe themselves in some base but not according to the patttern $6210001000$?
QUESTION [14 upvotes]: Call the first digit of a number digit $0$. The digit after that digit $1$, and so on and so forth.
In base $10$, the number $6210001000$ describes itself, because digit $0$ is $6$ and it has six $0$s. Digit $1$ is $2$ and it has $2$ $1s$. Digit $2$ is $1$ and there is only one $2$. There are none of the other numerals.
So it occurs to me that in duodecimal, you can do $821000001000$, which is $6073061476032$ in decimal. In base $40,$ you could do $Z21000...$ something like that, you get the idea. You could keep going, the only problem being not having enough symbols.
According to a graphic I saw on Facebook, $6210001000$ is the only such number in base $10$. I'm wondering: might there be some base in which a number describes itself like this but its first digit isn't 4 less than the base, followed by $21$ and a bunch of 0s with a 1 close to the end?

REPLY [2 votes]: From what I understand, the number 10,213,223 qualifies.  It has one zero, two ones, three twos, and two threes.  That's what led me to Google this idea of numbers that describe themselves.  I'm trying to see what other numbers qualify.