For \(b\geq 3\) there exist infinitely many base \(b\) \(k\)-Smith numbers (Q1567160)

For a base \(b\geq 3\) let \(S(x)\) denote the sum of the base \(b\) digits of \(x\) and let \(S_p(x)=\sum_{i=1}^r a_i S(p_i)\), where \(x=\prod_{i=1}^r p_i^{a_i}\) is the canonical form of \(x\). The number \(x\) is called a base \(b\) \(k\)-Smith number if \(x\) is composite and \(S_p(x)=kS(x)\). It was shown by \textit{W. L. McDaniel} [J. Number Theory 31, 91-98 (1989; Zbl 0673.10002)] that for \(b\geq 8\) there exist infinitely many \(b\) \(k\)-Smith numbers. Modifying the arguments of that paper, it is shown in the paper under review that this is true for any base \(b\geq 3\).