Practical numbers among the binomial coefficients

Paolo Leonetti*, Carlo Sanna

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)145-155
Number of pages11
JournalJournal of Number Theory
Publication statusPublished - Feb 2020


  • Binomial coefficient
  • Central binomial coefficient
  • Practical number

ASJC Scopus subject areas

  • Algebra and Number Theory


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