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