# self number

An integer $n$ that in a given base $b$ lacks a digitaddition generator.

Consider, for example, the integer 41 in base 10. It can be expressed as 34 + 3 + 4. For 42, however, there is no such digitaddition, hence it is a self number.

If $2|b$, all odd $$ will be self numbers.

Though self numbers form a small proportion of most ranges of $2b$ consecutive integers, there are infinitely many of them: The recurrence relation ${S}_{i}=(b-2){b}^{i-1}+{S}_{i-1}+(b-2)$ (with ${S}_{1}=b-1$ if $2|b$ and ${S}_{1}=b-2$ otherwise) will give an incomplete though infinite^{} list of self numbers.

Reference

Kaprekar, D. R. The Mathematics of New Self-Numbers. Devaiali, 1963: 19 - 20

Title | self number |
---|---|

Canonical name | SelfNumber |

Date of creation | 2013-03-22 15:56:15 |

Last modified on | 2013-03-22 15:56:15 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 6 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A63 |

Synonym | Columbian number |

Synonym | Colombian number |

Synonym | self-number |