# Keith number

Given a base $b$ integer

$$n=\sum _{i=1}^{k}{d}_{i}{b}^{i-1}$$ |

where ${d}_{1}$ is the least significant digit and ${d}_{k}$ is the most significant, construct the sequence ${a}_{1}={d}_{k},\mathrm{\dots}{a}_{k}={d}_{1}$, and for $m>k$,

$${a}_{m}=\sum _{i=1}^{k}{a}_{m-i}.$$ |

If there is an $x$ such that ${a}_{x}=n$, then $n$ is a Keith number or repfigit number.

In base 10, the first few Keith numbers below ten thousand are: 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909 (see A007629 in Sloane’s OEIS for a longer listing). 47 is a base 10 Keith number because it is contained the Fibonacci-like recurrence started from its base 10 digits: 4, 7, 11, 18, 29, 47, etc.

## References

- 1 M. Keith, “Repfigit Numbers” J. Rec. Math. 19 (1987), 41 - 42.

Title | Keith number |
---|---|

Canonical name | KeithNumber |

Date of creation | 2013-03-22 16:00:20 |

Last modified on | 2013-03-22 16:00:20 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A63 |

Synonym | repfigit number |