Explicit Bounds and Parallel Algorithms for Counting Multiply Gleeful Numbers

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.