Explicit Bounds and Parallel Algorithms for Counting Multiply Gleeful Numbers

TL;DR

This paper provides explicit bounds and develops parallel algorithms for counting and generating $k$-gleeful numbers, which are integers representable as sums of powers of consecutive primes, and explores their multiply-gleeful properties.

Contribution

It extends analytical bounds for $k$-gleeful numbers, introduces efficient parallel algorithms for their enumeration, and investigates the density and conjectures related to multiply-gleeful integers.

Findings

Explicit bounds on $k$-gleeful representations for $n \,\le\ x$.

Two new parallel algorithms for generating $k$-gleeful numbers.

Empirical data supporting heuristics on multiply-gleeful number density.

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 $k>1$. 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.