## Abstract

A practical number is a positive integer n such that every positive integer less than n can be written as a sum of distinct divisors of n. We prove that most of the binomial coefficients are practical numbers. Precisely, letting f(n) denote the number of binomial coefficients (nk), with 0≤k≤n, that are not practical numbers, we show that f(n)<n^{1−(log2−δ)/loglogn} for all integers n∈[3,x], but at most O_{γ}(x^{1−(δ−γ)/loglogx}) exceptions, for all x≥3 and 0<γ<δ<log2. Furthermore, we prove that the central binomial coefficient (2nn) is a practical number for all positive integers n≤x but at most O(x^{0.88097}) exceptions. We also pose some questions on this topic.

Original language | English |
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Pages (from-to) | 145-155 |

Number of pages | 11 |

Journal | Journal of Number Theory |

Volume | 207 |

DOIs | |

Publication status | Published - Feb 2020 |

## Keywords

- Binomial coefficient
- Central binomial coefficient
- Practical number

## ASJC Scopus subject areas

- Algebra and Number Theory