A Generalization of Collatz Functions and Jacobsthal Numbers
Ji Young Choi
Department of Mathematics
Shippensburg University of Pennsylvania
1871 Old Main Drive
Shippensburg, PA 17257
Let b ≥ 2 be an integer and g = b - 1. We consider a generalization of
the modified Collatz function: for any positive integer m, the
g-Collatz function fg divides m by g, if m is a multiple of g;
otherwise, the g-Collatz function fg is the least integer greater than
or equal to bm/g . Using this g-Collatz function, we extend the Collatz
problem, and we show that there are nontrivial cycles for some g.
Then we show how the function fg transforms the base-b representation
of positive integers, and we study the sequence of the b-ary
representation of integers
generated by the function fg, starting with
a b-ary string representing bN
for an arbitrary large integer N. We
show each b-ary string in the sequence has a repeating string, and the
number of occurrences of each digit in each shortest repeating string
generalizes Jacobsthal numbers.
Full version: pdf,
(Concerned with sequences
Received November 1 2017; revised versions received March 29 2018; May 11 2018.
Published in Journal of Integer Sequences, May 25 2018.
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