Journal of Integer Sequences, Vol. 21 (2018), Article 18.5.4

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.

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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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