Journal of Integer Sequences, Vol. 22 (2019), Article 19.5.5

Fixed Points of Augmented Generalized Happy Functions II: Oases and Mirages


Breeanne Baker Swart
Department of Mathematical Sciences
The Citadel
171 Moultrie St.
Charleston, SC 29409
USA

Susan Crook
Division of Mathematics, Engineering and Computer Science
Loras College
1450 Alta Vista St.
Dubuque, IA 52001
USA

Helen G. Grundman
Department of Mathematics
Bryn Mawr College
101 N. Merion Ave.
Bryn Mawr, PA 19010
USA

Laura Hall-Seelig
Department of Mathematics
Merrimack College
315 Turnpike Street
North Andover, MA 01845
USA

May Mei
Department of Mathematics and Computer Science
Denison University
100 West College Street
Granville, Ohio 43023
USA

Laurie Zack
Department of Mathematical Sciences
High Point University
One University Parkway
High Point, NC 27268
USA

Abstract:

An augmented generalized happy function S[c,b] maps a positive integer to the sum of the squares of its base b digits plus c. For b ≥ 2 and k ∈ Z+, a k-desert base b is a set of k consecutive non-negative integers c for each of which S[c,b] has no fixed points. In this paper, we examine a complementary notion, a k-oasis base b, which we define to be a set of k consecutive non-negative integers c for each of which S[c,b] has a fixed point. In particular, after proving some basic properties of oases base b, we compute bounds on the lengths of oases base b and compute the minimal examples of maximal length oases base b for small values of b.

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(Concerned with sequence A007770.)

Received February 8 2019; revised version received July 31 2019. Published in Journal of Integer Sequences, August 23 2019.

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