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−(log2−δ)/loglogn for all integers n∈[3,x], but at most Oγ(x1−(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(x0.88097) exceptions. We also pose some questions on this topic.
ASJC Scopus subject areas
- !!Algebra and Number Theory