# exponential factorial

Given a positive integer $n$, the ”power tower” ${n}^{{(n-1)}^{(n-2)\mathrm{\dots}}}$ is the exponential factorial of $n$. The recurrence relation is ${a}_{1}=1$, ${a}_{n}={n}^{{a}_{n-1}}$ for $n>1$.

So for example, $9={3}^{{2}^{1}}$, $262144={4}^{{3}^{{2}^{1}}}$. The exponential factorial for 5 has almost two hundred thousand base 10 digits. The ones that are small enough are listed in sequence^{} A049384 of Sloane’s OEIS.

The sum of the reciprocals of the exponential factorials is a Liouville number^{}.

$$\sum _{i=1}^{\mathrm{\infty}}\frac{1}{{a}_{i}}\approx 1.6111149258083767361111111$$ |

Title | exponential factorial |
---|---|

Canonical name | ExponentialFactorial |

Date of creation | 2013-03-22 16:01:38 |

Last modified on | 2013-03-22 16:01:38 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 7 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 05A10 |

Related topic | Factorial^{} |