Explicit Bounds and Parallel Algorithms for Counting Multiply Gleeful Numbers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915385671942144 |
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| author | Moore, Sara Sorenson, Jonathan P. |
| author_facet | Moore, Sara Sorenson, Jonathan P. |
| contents | Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful. In this paper, we present some new results on $k$-gleeful numbers for $k>1$.
First, we extend previous analytical work. For given values of $x$ and $k$, we give explicit upper and lower bounds on the number of $k$-gleeful representations of integers $n\le x$.
Second, we describe and analyze two new, efficient parallel algorithms, one theoretical and one practical, to generate all $k$-gleeful representations up to a bound $x$.
Third, we study integers that are multiply gleeful, that is, integers with more than one representation as a sum of powers of consecutive primes, including both the same or different values of $k$. We give a simple heuristic model for estimating the density of multiply-gleeful numbers, we present empirical data in support of our heuristics, and offer some new conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Bounds and Parallel Algorithms for Counting Multiply Gleeful Numbers Moore, Sara Sorenson, Jonathan P. Number Theory Data Structures and Algorithms 11A41, 11Y16, 68Q25 F.2.1 Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful. In this paper, we present some new results on $k$-gleeful numbers for $k>1$. First, we extend previous analytical work. For given values of $x$ and $k$, we give explicit upper and lower bounds on the number of $k$-gleeful representations of integers $n\le x$. Second, we describe and analyze two new, efficient parallel algorithms, one theoretical and one practical, to generate all $k$-gleeful representations up to a bound $x$. Third, we study integers that are multiply gleeful, that is, integers with more than one representation as a sum of powers of consecutive primes, including both the same or different values of $k$. We give a simple heuristic model for estimating the density of multiply-gleeful numbers, we present empirical data in support of our heuristics, and offer some new conjectures. |
| title | Explicit Bounds and Parallel Algorithms for Counting Multiply Gleeful Numbers |
| topic | Number Theory Data Structures and Algorithms 11A41, 11Y16, 68Q25 F.2.1 |
| url | https://arxiv.org/abs/2507.09012 |