# sphenic number

Given three primes $$, the composite integer $pqr$ is a sphenic number^{}. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, $\mathrm{\dots}$ listed in A007304 of Sloane’s OEIS.

The divisors^{} of a sphenic number therefore are $1,p,q,r,pq,pr,qr,pqr$. Furthermore, $\mu (pqr)={(-1)}^{3}$ (where $\mu $ is the Möbius function^{}), $\tau (pqr)=8$ (where $\tau $ is the divisor function^{}) and $\mathrm{\Omega}(pqr)=\omega (pqr)=3$ (where $\mathrm{\Omega}$ and $\omega $ are the number of (nondistinct) prime factors^{} function (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction) and the number of distinct prime factors function, respectively).

The largest known sphenic number at any time is usually the product of the three largest known Mersenne primes^{}.

Title | sphenic number |
---|---|

Canonical name | SphenicNumber |

Date of creation | 2013-03-22 16:10:33 |

Last modified on | 2013-03-22 16:10:33 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 7 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A05 |