A Curious Bijection on Natural Numbers
B. J. Venkatachala
MO Cell, HBCSE(TIFR)
Department of Mathematics
Indian Institute of Science
We give a greedy algorithm for describing an enumeration of the set of all
natural numbers such that the sum of the first n terms of the
sequence is divisible by n for each natural number n. We show
that this leads to a bijection f of the set of all natural
numbers onto itself that has some nice properties. We also show
that the average function of the first n terms of the
sequence satisfies a functional equation which completely describes
all the properties of the function f.
In particular, f turns out to
be an involution on the set of all natural numbers.
Full version: pdf,
(Concerned with sequence
Received June 16 2009;
revised version received November 11 2009.
Published in Journal of Integer Sequences, November 16 2009.
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