Rocky Mountain Journal of Mathematics

Augmented generalized happy functions

B. Baker Swart, K.A. Beck, S. Crook, C. Eubanks-Turner, H.G. Grundman, M. Mei, and L. Zack

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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~$\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.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 2 (2017), 403-417.

Dates
First available in Project Euclid: 18 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1492502542

Digital Object Identifier
doi:10.1216/RMJ-2017-47-2-403

Mathematical Reviews number (MathSciNet)
MR3635366

Zentralblatt MATH identifier
1380.11005

Subjects
Primary: 11A63: Radix representation; digital problems {For metric results, see 11K16}

Keywords
Happy numbers iteration integer functions

Citation

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


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References

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