Augmented generalized happy functions
SMC Affiliated Work
1
Status
Faculty
School
School of Science
Department
Math/Computer Science
Document Type
Article
Publication Date
2017
Publication / Conference / Sponsorship
Rocky Mountain Journal of Mathematics
Description/Abstract
An augmented generalized happy function, S[c,b]S[c,b] maps a positive integer to the sum of the squares of its base bb digits and a non-negative integer~cc. A positive integer uu is in a \textit {cycle} of S[c,b]S[c,b] if, for some positive integer~kk, S[c,b]k(u)=uS[c,b]k(u)=u, and, for positive integers vv and ww, vv is ww-\textit {attracted} for S[c,b]S[c,b] if, for some non-negative integer~ℓℓ, S[c,b]ℓ(v)=wS[c,b]ℓ(v)=w. In this paper, we prove that, for each c≥0c≥0 and b≥2b≥2, and for any uu in a cycle of S[c,b]S[c,b]: (1)~if bb is even, then there exist arbitrarily long sequences of consecutive uu-attracted integers, and (2)~if bb is odd, then there exist arbitrarily long sequences of 2-consecutive uu-attracted integers.
Keywords
Happy numbers, iteration, integer functions
Scholarly
yes
DOI
10.1216/RMJ-2017-47-2-403
Volume
47
Issue
2
First Page
403
Last Page
417
Disciplines
Mathematics
Rights
Open Access. Author manuscript (arXiv)
Original Citation
Kristen Beck (Mathematics and Computer Science): Swart, B. Baker; Beck, K.A.; Crook, S.; Eubanks-Turner, C.; Grundman, H.G.; Mei, M.; Zack, L. Augmented generalized happy functions. Rocky Mountain J. Math. 47 (2017), no. 2, 403--417. doi:10.1216/RMJ-2017-47-2-403. https://projecteuclid.org/euclid.rmjm/1492502542.
Repository Citation
Beck, Kristen and author(s), additional. Augmented generalized happy functions (2017). Rocky Mountain Journal of Mathematics. 47 (2), 403-417. 10.1216/RMJ-2017-47-2-403 [article]. https://digitalcommons.stmarys-ca.edu/school-science-faculty-works/36
