Fixed points of augmented generalized happy functions

Breeanne Baker Swart; Kristen A Beck; Susan Crook; Christina Eubanks-Turner; Helen G Grundman; May Mei; Laurie Zack

doi:10.1216/RMJ-2018-48-1-47

Fixed points of augmented generalized happy functions

Breeanne Baker Swart
, Kristen A Beck
, Susan Crook
, Christina Eubanks-Turner
, Helen G Grundman
, May Mei
, Laurie Zack

Mathematics and Statistics

The Citadel - The Military College of South Carolina
Saint Mary's College of California
Loras College
Loyola Marymount University
Bryn Mawr College
Denison University
High Point University

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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. In this paper, we study various properties of the fixed points of S[c,b]; count the number of fixed points of S[c,b] for b ≥ 2 and 0 < c < 3b − 3; and prove that, for each b ≥ 2, there exist arbitrarily many consecutive values of c for which S[c,b] has no fixed point.

Original language	English
Pages (from-to)	47-58
Number of pages	12
Journal	Rocky Mountain Journal of Mathematics
Volume	48
Issue number	1
DOIs	https://doi.org/10.1216/RMJ-2018-48-1-47
State	Published - Jan 1 2018

Keywords

Fixed points.
Happy numbers
Integer functions
Iteration

Access to Document

10.1216/RMJ-2018-48-1-47

Cite this

@article{9cecafa228da41cf826f348cacda1534,

title = "Fixed points of augmented generalized happy functions",

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. In this paper, we study various properties of the fixed points of S[c,b]; count the number of fixed points of S[c,b] for b ≥ 2 and 0 < c < 3b − 3; and prove that, for each b ≥ 2, there exist arbitrarily many consecutive values of c for which S[c,b] has no fixed point.",

keywords = "Fixed points., Happy numbers, Integer functions, Iteration",

author = "Swart, \{Breeanne Baker\} and Beck, \{Kristen A\} and Susan Crook and Christina Eubanks-Turner and Grundman, \{Helen G\} and May Mei and Laurie Zack",

year = "2018",

month = jan,

day = "1",

doi = "10.1216/RMJ-2018-48-1-47",

language = "English",

volume = "48",

pages = "47--58",

journal = "Rocky Mountain Journal of Mathematics",

issn = "0035-7596",

publisher = "Rocky Mountain Mathematics ConsortiumPO Box 871804TempeAZ 85287-1804",

number = "1",

}

TY - JOUR

T1 - Fixed points of augmented generalized happy functions

AU - Swart, Breeanne Baker

AU - Beck, Kristen A

AU - Crook, Susan

AU - Eubanks-Turner, Christina

AU - Grundman, Helen G

AU - Mei, May

AU - Zack, Laurie

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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. In this paper, we study various properties of the fixed points of S[c,b]; count the number of fixed points of S[c,b] for b ≥ 2 and 0 < c < 3b − 3; and prove that, for each b ≥ 2, there exist arbitrarily many consecutive values of c for which S[c,b] has no fixed point.

AB - 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. In this paper, we study various properties of the fixed points of S[c,b]; count the number of fixed points of S[c,b] for b ≥ 2 and 0 < c < 3b − 3; and prove that, for each b ≥ 2, there exist arbitrarily many consecutive values of c for which S[c,b] has no fixed point.

KW - Fixed points.

KW - Happy numbers

KW - Integer functions

KW - Iteration

UR - https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85046812935&origin=inward

UR - https://www.scopus.com/inward/citedby.uri?partnerID=HzOxMe3b&scp=85046812935&origin=inward

U2 - 10.1216/RMJ-2018-48-1-47

DO - 10.1216/RMJ-2018-48-1-47

M3 - Article

SN - 0035-7596

VL - 48

SP - 47

EP - 58

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

IS - 1

ER -

Fixed points of augmented generalized happy functions

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this