## School of Science Faculty Works

#### Title

Augmented generalized happy functions

1

Faculty

#### School

School of Science

#### Department

Math/Computer Science

Article

2017

#### Publication Title

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

yes

#### DOI

10.1216/RMJ-2017-47-2-403

47

2

403

417

Mathematics

#### 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.

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