##
**Gaps between consecutive divisors of factorials.**
*(English)*
Zbl 0790.11007

The set of all divisors of \(n!\), ordered according to increasing magnitude, is considered, and an upper bound on the gaps between consecutive ones is obtained. We are especially interested in the divisors nearest \(\sqrt{n!}\) and obtain a lower bound on their distance.

Reviewer: J.E.Harmse (Austin)

### Keywords:

consecutive divisors of factorials; gaps between divisors; divisor sequences; upper bounds; lower bounds### Online Encyclopedia of Integer Sequences:

Number of divisors of n!.Number of divisors of n! which are also differences between consecutive divisors of n! (ordered by size).

### References:

[1] | [1] and , On the equation P(x) = n! and a question of Erdös, J. of Number Theory, 42 (1992), 189-193. · Zbl 0762.11010 |

[2] | [2] , Some problems and results in number theory, Number Theory and Combinatorics, Japan 1984, World Scientific, Singapore, 1985, 65-87. · Zbl 0603.10001 |

[3] | [3] , Some problems and results on additive and multiplicative number theory, Analytic Number Theory, (Philadelphia, 1980), Springer-Verlag Lecture Notes, 899 (1981), 171-182. · Zbl 0472.10002 |

[4] | [4] , Some solved and unsolved problems of mine in number theory, Topics in Analytic Number Theory, University of Texas Press, Austin, 1985, 59-75. · Zbl 0596.10001 |

[5] | [5] , Personal communication. |

[6] | [6] and , Divisors, Cambridge University Press, Cambridge, 1988. · Zbl 0653.10001 |

[7] | [7] , Substitution Dynamical Systems - Spectral Analysis, Springer-Verlag Lecture Notes, 1294, Berlin, 1987. · Zbl 0642.28013 |

[8] | [8] , Sur un problème extrémal en arithmétique, Ann. Inst. Fourier, Grenoble, 37-2 (1987), 1-18. · Zbl 0622.10030 |

[9] | [9] , Integers with consecutive divisors in small ratio, J. of Number Theory, 19 (1984), 233-238. · Zbl 0543.10031 |

[10] | [10] , Limit theorems for divisor distributions, Proc. Amer. Math. Soc., 95 (1985), 505-511. · Zbl 0609.10041 |

[11] | [11] , The distribution of divisors of N!, Acta Arith., 50 (1988), 203-209. · Zbl 0647.10038 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.