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
México
Abstract:
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.
Full version: pdf,
dvi,
ps,
latex
(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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