Abstract
Let be a positive integer and be the sum of the squares of its decimal digits. When there exists a positive integer such that the -th iterate of on is 1, i.e., , then is called a happy number. The notion of happy numbers has been generalized to different bases, different powers and even negative bases. In this article we consider generalizations to fractional number bases. Let be the sum of the -th powers of the digits of base . Let be the smallest nonnegative integer for which there exists a positive integer satisfying . We prove that such a , called the height of , exists for all , and that, if or , then can be arbitrarily large.
Citation
Enrique Treviño. Mikita Zhylinski. "On generalizing happy numbers to fractional-base number systems." Involve 12 (7) 1143 - 1151, 2019. https://doi.org/10.2140/involve.2019.12.1143
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