### Abstract

Let f(n) denote the number of factorizations of the natural number n into factors larger than 1 where the order of the factors does not count. We say n is "highly factorable" if f(m)<f(n) for all m < n. We prove that f(n)=n·L(n)^{-1+0(1)} for n highly factorable, where L(n)=exp{log n logloglog n loglog n}. This result corrects the 1926 paper of Oppenheim where it is asserted that f(n)=n·L(n)^{-2+0(1)}. Some results on the multiplicative structure of highly factorable numbers are proved and a table of them up to 10^{9} is provided. Of independent interest, a new lower bound is established for the function Ψ(x, y), the number of n≤x free of prime factors exceeding y.

Original language | English |
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Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Journal of Number Theory |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 1983 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Number Theory*,

*17*(1), 1-28. https://doi.org/10.1016/0022-314X(83)90002-1