self-descriptive number

A self-descriptive number $n$ in base $b$ is an integer such that each base $b$ digit

$$d_x=\sum _{d_i=x}1$$

where each $d_i$ is a digit of $n$, $i$ is a very simple, standard iterator operating in the range , and $x$ is a position of a digit; thus $n$ “describes” itself.

For example, the integer 6210001000 written in base 10. It has six instances of the digit 0, two instances of the digit 1, a single instance of the digit 2, a single instance of the digit 6 and no instances of any other base 10 digits.

Base 4 might be the only base with two self-descriptive numbers, ${1210}_4$ and ${2020}_4$. From base 7 onwards, every base $b$ has at least one self-descriptive number of the form ${(b-4)}^{b-1}+2b^{b-2}+b^{b-3}+b^4$. It has been proven that 6210001000 is the only self-descriptive number in base 10, but it’s not known if any higher bases have any self-descriptive numbers of any other form.

Sequence A108551 of the OEIS lists self-descriptive numbers from quartal to hexadecimal.

Title	self-descriptive number
Canonical name	SelfdescriptiveNumber
Date of creation	2013-03-22 15:53:27
Last modified on	2013-03-22 15:53:27
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	9
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A63
Synonym	self descriptive number