Counting Keith numbers

Klazar, Martin ; Luca, Florian

Journal of integer sequences, Tome 10 (2007) no. 2

Résumé

A Keith number is a positive integer $N$ with decimal representation $a_{1} a_{2} \dots a_{n}$ such that $n >= 2$ and $N$ appears in the sequence $(K_{m})_{m >= 1}$ given by the recurrence $K_{1} = a_{1}, \dots , K_{n} = a_{n}$ and $K_{m} = K_{m-1} + K_{m-2} + \dots + K_{m-n}$ for $m > n$. We prove that there are only finitely many Keith numbers using only one decimal digit (i.e., $a_{1}= a_{2}= \dots = a_{n}$), and that the set of Keith numbers is of asymptotic density zero.

Détail

Classification : 11B39, 11A63
Keywords: keith number, density, generalized Fibonacci recurrence