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

Counting Keith Numbers

Martin Klazar
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


A Keith number is a positive integer N with decimal representation a1 a2 ... an such that n >= 2 and N appears in the sequence (Km)m >= 1 given by the recurrence K1 = a1, ... , Kn = an and Km = Km-1 + Km-2 + ... + Km-n for m > n. We prove that there are only finitely many Keith numbers using only one decimal digit (i.e., a1= a2= ... = an), and that the set of Keith numbers is of asymptotic density zero.

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(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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