2025/07/11 by Sara Moore, Moore, Sara, Jonathan Sorenson +1
Computer Science · #11A41 #11Y16 #68Q25 #Data Structures and Algorithms (cs.DS) #F.2.1 #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.2507.09012
openalex publication_date 2025/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k≥ 1 be an integer. A positive integer n is k-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.