Open Access
2017 Augmented generalized happy functions
B. Baker Swart, K.A. Beck, S. Crook, C. Eubanks-Turner, H.G. Grundman, M. Mei, L. Zack
Rocky Mountain J. Math. 47(2): 403-417 (2017). DOI: 10.1216/RMJ-2017-47-2-403

Abstract

An augmented generalized happy function, S[c,b] maps a positive integer to the sum of the squares of its base b digits and a non-negative integer~c. A positive integer u is in a \textit {cycle} of S[c,b] if, for some positive integer~k, S[c,b]k(u)=u, and, for positive integers v and w, v is w-\textit {attracted} for S[c,b] if, for some non-negative integer~, S[c,b](v)=w. In this paper, we prove that, for each c0 and b2, and for any u in a cycle of S[c,b]: (1)~if b is even, then there exist arbitrarily long sequences of consecutive u-attracted integers, and (2)~if b is odd, then there exist arbitrarily long sequences of 2-consecutive u-attracted integers.

Citation

Download Citation

B. Baker Swart. K.A. Beck. S. Crook. C. Eubanks-Turner. H.G. Grundman. M. Mei. L. Zack. "Augmented generalized happy functions." Rocky Mountain J. Math. 47 (2) 403 - 417, 2017. https://doi.org/10.1216/RMJ-2017-47-2-403

Information

Published: 2017
First available in Project Euclid: 18 April 2017

zbMATH: 1380.11005
MathSciNet: MR3635366
Digital Object Identifier: 10.1216/RMJ-2017-47-2-403

Subjects:
Primary: 11A63

Keywords: happy numbers , integer functions , iteration

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

Vol.47 • No. 2 • 2017
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