# primary pseudoperfect number

Given an integer $n$ with $\omega (n)$ distinct prime factors ${p}_{i}$ (where $\omega $ is number of distinct prime factors function), if the equality

$$\frac{1}{n}+\sum _{i=1}^{\omega (n)}\frac{1}{{p}_{i}}=1$$ |

holds true, then $n$ is a *primary pseudoperfect number*. Equivalently,

$$1+\sum _{i=1}^{\omega (n)}\frac{n}{{p}_{i}}=n$$ |

if $n$ is a primary pseudoperfect number.

The first few primary pseudoperfect numbers are 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086, the first four of these being each one less than the first four terms of Sylvester’s sequence; these are listed in A054377 of Sloane’s OEIS. Presently it’s not known whether there are any odd primary pseudoperfect numbers.

Title | primary pseudoperfect number |
---|---|

Canonical name | PrimaryPseudoperfectNumber |

Date of creation | 2013-03-22 16:17:40 |

Last modified on | 2013-03-22 16:17:40 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 6 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11D85 |

Related topic | GiugaNumber |