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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


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~$\ell $, ${S_{[c,b]}} ^\ell (v) = w$. In this paper, we prove that, for each $c\geq 0$ and $b \geq 2$, 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.


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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.


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

Primary: 11A63

Keywords: happy numbers , integer functions , iteration

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium


Vol.47 • No. 2 • 2017
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