Almost Repdigits in $ k-$generalized Lucas Sequences

TL;DR

This paper investigates which terms in k-generalized Lucas sequences are almost repdigits, including powers of 10, and aims to classify all such terms.

Contribution

It characterizes all k-generalized Lucas sequence terms that are almost repdigits, extending previous work to a broader class of sequences and including powers of 10.

Findings

Identifies all sequence terms that are powers of 10.

Classifies all almost repdigit terms in k-generalized Lucas sequences.

Provides explicit solutions for specific cases.

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.