self-descriptive number

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


where each di is a digit of n, i is a very simple, standard iterator operating in the range -1<i<b, 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, 12104 and 20204. From base 7 onwards, every base b has at least one self-descriptive number of the form (b-4)b-1+2bb-2+bb-3+b4. 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