Almost Repdigits in $ k-$generalized Lucas Sequences

Abstract

Let $ k \geq 2 $ and $ ( L_{n}^{(k)} )_{n \geq 2-k} $ be the $k-$generalized Lucas sequence with initial condition $ L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)}=0 ,$ $ L_{0}^{(k,}=2,$ $ L_{1}^{(k)}=1$ and each term afterwards is the sum of the $ k $ preceding terms. A positive integer is an almost repdigit if its digits are all equal except for at most one digit. In this paper, we work on the problem of determining all terms of $k-$generalized Lucas sequences which are almost repdigits. In particular, we find all $k-$generalized Lucas numbers which are powers of $10$ as a special case of almost repdigits.