A tree approach to the happy function
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909353989111808 |
|---|---|
| author | Goedhart, Eva G. Gurtas, Yusuf Harris, Pamela E. |
| author_facet | Goedhart, Eva G. Gurtas, Yusuf Harris, Pamela E. |
| contents | In this article, we present a method to construct $e$-power $b$-happy numbers of any height. Using this method, we construct a tree that encodes these happy numbers, their heights, and their ancestry--relation to other happy numbers. For fixed power $e$ and base $b$, we consider happy numbers with at most $k$ digits and we give a formula for the cardinality of the preimage of a single iteration of the happy function. We show that these happy numbers arise naturally as children of a given vertex in the tree. We conclude by applying this technique to $e$-power $b$-unhappy numbers of a given height. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13990 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A tree approach to the happy function Goedhart, Eva G. Gurtas, Yusuf Harris, Pamela E. Number Theory 11A63 In this article, we present a method to construct $e$-power $b$-happy numbers of any height. Using this method, we construct a tree that encodes these happy numbers, their heights, and their ancestry--relation to other happy numbers. For fixed power $e$ and base $b$, we consider happy numbers with at most $k$ digits and we give a formula for the cardinality of the preimage of a single iteration of the happy function. We show that these happy numbers arise naturally as children of a given vertex in the tree. We conclude by applying this technique to $e$-power $b$-unhappy numbers of a given height. |
| title | A tree approach to the happy function |
| topic | Number Theory 11A63 |
| url | https://arxiv.org/abs/2410.13990 |