# 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 ${(b-4)}^{b-1}+2{b}^{b-2}+{b}^{b-3}+{b}^{4}$. 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 |