Multiperfect numbers with identical digits (Q616426)

Let \(g\geq 2\). A natural number \(N\) is called repdigit in base \(g\) if all digits in its base \(g\) expansion are equal. Equivalently, \[ \exists\, m\geq 1,\quad D\in\{1,2,\ldots,g-1\},\quad N=D\frac{g^m-1}{g-1}. \] A natural number \(N\) is called multiperfect if \(\sigma(N)\) is a proper multiple of \(N\), \(\sigma\) the sum-of-divisors function. The authors prove that there are only finitely many repdigit multiperfect numbers \(N\) in base \(g\). There is a computable upper bound on the number of such \(N\). For \(g=10\) the only multiperfect repdigit is \(N=6\).