The combinatorial method to compute the sum of the powers of primes

TL;DR

This paper generalizes combinatorial algorithms originally designed for counting primes to efficiently compute the sum of prime powers up to a given limit, including detailed algorithms and implementation insights.

Contribution

It introduces a novel extension of combinatorial methods to calculate sums of prime powers, broadening the scope of prime counting algorithms.

Findings

Developed algorithms for summing prime powers efficiently.

Provided detailed implementation and algorithmic analysis.

Extended combinatorial prime counting techniques to new mathematical functions.

Abstract

We will generalize the combinatorial algorithms for computing $\pi(x)$ to compute sums ${F(x) = \sum_{p \leq x} p^k}$ for $k \in \mathbb{Z}_{\geq 0}$. The detailed exposition of algorithms is included along with implementation details.