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A Generalization of Collatz Functions and Jacobsthal Numbers
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Ji Young Choi

Department of Mathematics

Shippensburg University of Pennsylvania

1871 Old Main Drive

Shippensburg, PA 17257

USA

**Abstract:**

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 *f*_{g} divides *m* by *g*, if *m* is a multiple of *g*;
otherwise, the *g*-Collatz function *f*_{g} 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 *f*_{g} transforms the base-*b* representation
of positive integers, and we study the sequence of the *b*-ary
representation of integers
generated by the function *f*_{g}, starting with
a *b*-ary string representing *b*^{N}
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.

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(Concerned with sequences
A000975
A001045
A005578
A015331
A015518
A015521
A015540
A054878
A078008
A109499
A109500
A109501
A122983.)

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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