Practical numbers among the binomial coefficients (Q2329268)

A practical number is a positive integer \(n\), such that every positive integer less then \(n\) is a sum of distinct divisors of \(n\). The authors study binomial coefficients, that are also practical numbers. Let \(f(n)\) the number of \(\binom{n}{k}\), \(0\le k\le n\) that are practical. They prove \[f(n)<n^{1-\frac{c}{\log\log n}}\] for all \(3\le n\le x\) with at most \(O(x^{1-\frac{n}{\log\log n}})\) exceptions. Furthermore they show that the central binomial coefficient \(\binom{2n}{n}\) is practical for all \(n\le x\) with at most \(O(x^{0,88097})\) exceptions. They suppose, that there are only finite many. The proofs use many results of Melfi, Fine, Kummer and Okamoto.