Divisibility properties of hyperharmonic numbers (Q1705202)

The \(n\)th hyperharmonic number of order \(r\) is defined recursively by \(h_n^{(r)}=\sum_{k=1}^n h_k^{(r-1)}\), where \(h_n^{(1)}=\sum_{k=1}^n k^{-1}\) is the \(n\)th harmonic number. The authors prove a series of results for hyperharmonic numbers in this long paper. For instance, a generalization of Wolstenholme's theorem for hyperharmonic numbers is proved. Using combinatorial arguments they analyze hyperharmonic numbers which are not divisible by a prime. In particular, this yields that all hyperharmonic numbers in their reduced fractional form are odd. Given an odd prime \(p\), they further prove that card\(\{(n,r)\in {\mathbb N} : 1\leq n,r\leq x, h_n^{(r)}\equiv 0\pmod{p}\}\gg_p x^{\tau(p)}\) with explicitly given \(\tau(p)\). Contrary to an open problem whether there are only finitely many harmonic numbers \(h_n\) which are divisible by given prime \(p\), they show there are infinitely many hyperharmonic numbers \(h_n^{(r)}\) which are divisible by \(p\). More precisely, if \(J_p^{(r)}=\{n, \in {\mathbb N} : h_n^{(r)}\equiv 0\pmod{p}\}\) and \(p\geq r\geq2\), \(p\) odd prime, then for the counting function of \(J_p^{(r)}\) we have \(J_p^{(r)}(x)\gg_p \log x\). For the rest of results we refer the interested reader to the original paper which ends with two conjectures.