Document Type

Article

Publication Date

7-11-2025

Publication Title

arxiv.org

First Page

1

Last Page

17

DOI

https://doi.org/10.48550/arXiv.2507.09012

Additional Publication URL

https://arxiv.org/abs/2507.09012

Abstract

Let k ≥ 1 be an integer. A positive integer n is k-\textit{gleeful} if n can be represented as the sum of kth powers of consecutive primes. For example, 35=23+33 is a 3-gleeful number, and 195=52+72+112 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 ≤ 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.

Rights

This is a pre-print version of this article. The version of record is available at Cambridge University Press.

NOTE: this version of the article is pending revision and may not reflect the changes made in the final, peer-reviewed version.

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