Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.2 |

Department of Applied Mathematics and

Institute for Theoretical Computer Science (ITI)

Faculty of Mathematics and Physics

Charles University

Malostranské nám. 25

11800 Praha

Czech Republic

Florian Luca

Instituto de Matemáticas

Universidad Nacional Autonoma de México

C.P. 58089

Morelia, Michoacán

México

**Abstract:**

A Keith number is a positive integer *N* with decimal
representation
*a*_{1}
*a*_{2} ...
*a*_{n}
such that *n* >= 2 and *N*
appears in the sequence
(*K*_{m})_{m >= 1}
given by the
recurrence *K*_{1} = *a*_{1}, ... ,
*K*_{n} = *a*_{n} and
*K*_{m} =
*K*_{m-1} +
*K*_{m-2} + ... +
*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}= ... =
*a*_{n}),
and that the set of Keith numbers is of asymptotic density zero.

(Concerned with sequence A007629.)

Received September 21 2006;
revised version received January 16 2007.
Published in *Journal of Integer Sequences* January 17 2007.

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