{"ID":2895865,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.09012","arxiv_id":"2507.09012","title":"Explicit Bounds and Parallel Algorithms for Counting Multiply Gleeful Numbers","abstract":"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 $k\u003e1$. 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.","short_abstract":"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 $k\u003e1$. Fi...","url_abs":"https://arxiv.org/abs/2507.09012","url_pdf":"https://arxiv.org/pdf/2507.09012v1","authors":"[\"Sara Moore\",\"Jonathan P. Sorenson\"]","published":"2025-07-11T20:36:49Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"cs.DS\"]","methods":"[]","has_code":false}