Let \(H_n:=1+\frac 12+\ldots+\frac 1n \) be the harmonic numbers. For a positive integer \(r\) the hyperharmonic numbers of order \(r\) are defined by \[ H_n^{(1)}:=H_n,H_n^{(r)}= \sum_{k=1}^nH_k^{(r-1)}, r>1. \] The authors prove: For any \(s\in(1,2)\) there is a prime number \(P_0\) such that for any integers \(r\) and \(n\) with \(5\leq r\leq (2-s)P_0+2\) and \(n\geq P_0\) the number \(H_n^{(r)}\) is not an integer. The number \(H_n^{(r)}\) is not an integer for \(n\geq 2\) and \(5\leq r\leq 25\), for \(n\geq 2.010.881\) and \(5\leq r\leq 2.010.761\), for \(n\geq 10.726.905.041\) and \(5\leq r\leq 10.726.904.664\).