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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.09863 |
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| _version_ | 1866910605126926336 |
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| author | Fox, N. Bradley Fox, Nathan H. Grundman, Helen G. Lynn, Rachel Namoijam, Changningphaabi Vanderschoot, Mary |
| author_facet | Fox, N. Bradley Fox, Nathan H. Grundman, Helen G. Lynn, Rachel Namoijam, Changningphaabi Vanderschoot, Mary |
| contents | For a base $b \geq 2$, the $b$-elated function, $E_{2,b}$, maps a positive integer written in base $b$ to the product of its leading digit and the sum of the squares of its digits. A $b$-elated number is a positive integer that maps to $1$ under iteration of $E_{2,b}$. The height of a $b$-elated number is the number of iterations required to map it to $1$. We determine the fixed points and cycles of $E_{2,b}$ and prove a range of results concerning sequences of $b$-elated numbers and $b$-elated numbers of minimal heights. Although the $b$-elated function is closely related to the $b$-happy function, the behaviors of the two are notably different, as demonstrated by the results in this work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elated Numbers Fox, N. Bradley Fox, Nathan H. Grundman, Helen G. Lynn, Rachel Namoijam, Changningphaabi Vanderschoot, Mary Number Theory 11A63 For a base $b \geq 2$, the $b$-elated function, $E_{2,b}$, maps a positive integer written in base $b$ to the product of its leading digit and the sum of the squares of its digits. A $b$-elated number is a positive integer that maps to $1$ under iteration of $E_{2,b}$. The height of a $b$-elated number is the number of iterations required to map it to $1$. We determine the fixed points and cycles of $E_{2,b}$ and prove a range of results concerning sequences of $b$-elated numbers and $b$-elated numbers of minimal heights. Although the $b$-elated function is closely related to the $b$-happy function, the behaviors of the two are notably different, as demonstrated by the results in this work. |
| title | Elated Numbers |
| topic | Number Theory 11A63 |
| url | https://arxiv.org/abs/2409.09863 |