## Rocky Mountain Journal of Mathematics

### 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 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

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

#### 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

#### References

• E. El-Sedy and S. Sisek, On happy numbers, Rocky Mountain J. Math. 30 (2000), 565–570.
• H.G. Grundman and E.A. Teeple, Generalized happy numbers, Fibonacci Quart. 39 (2001), 462–466.
• ––––, Sequences of consecutive happy numbers, Rocky Mountain J. Math. 37 (2007), 1905–1916.
• ––––, Sequences of generalized happy numbers with small bases, J. Integer Sequences 10 (2007), article 07.1.8.
• Richard K. Guy, Unsolved problems in number theory, Third edition, in Problem books in mathematics, Springer-Verlag, New York, 2004.
• Ross Honsberger, Ingenuity in mathematics, New Math. Library 23, Random House, Inc., New York, 1970.