Practical numbers among the binomial coefficients

Paolo Leonetti*, Carlo Sanna

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel

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)<n1−(log⁡2−δ)/log⁡log⁡n for all integers n∈[3,x], but at most Oγ(x1−(δ−γ)/log⁡log⁡x) exceptions, for all x≥3 and 0<γ<δ<log⁡2. Furthermore, we prove that the central binomial coefficient (2nn) is a practical number for all positive integers n≤x but at most O(x0.88097) exceptions. We also pose some questions on this topic.

Originalspracheenglisch
Seiten (von - bis)145-155
Seitenumfang11
FachzeitschriftJournal of Number Theory
Jahrgang207
DOIs
PublikationsstatusVeröffentlicht - Feb 2020

ASJC Scopus subject areas

  • !!Algebra and Number Theory

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