The function \(d_k (n)\) at consecutive integers (Q1976632)

Let \(d_k(n)\) denote the \(k\)-fold iterated divisor function \((k\geq 2)\). It is proved that for sufficiently large \(x\), \(d_k(n)= d_k(n+1)\) holds for \(\gg x(\log\log x)^{-3}\) integers \(n\leq x\). This improves earlier results of \textit{D. R. Heath-Brown} [Mathematika 31, 141-149 (1984; Zbl 0529.10040)] and \textit{A. Hildebrand} [Pac. J. Math. 129, 307-319 (1987; Zbl 0591.10035)]. Here the author combines the two methods and extends them properly.