Large strings of consecutive smooth integers

TL;DR

This paper improves an existing algorithm to compute the minimal gap size function for integers with large prime factors, enabling the calculation of new values of this function for larger parameters.

Contribution

It introduces an enhanced algorithm that allows for the computation of the Erdős-related gap function for larger values of k, advancing previous methods.

Findings

Computed new values of f(k) for larger k

Improved algorithm efficiency

Extended understanding of prime factor gaps

Abstract

In this note we improve an algorithm from a recent paper by Bauer and Bennett for computing a function of Erd\"os that measures the minimal gap size $f(k)$ in the sequence of integers at least one of whose prime factors exceeds $k$. This allows us to compute values of $f(k)$ for larger $k$ and obtain new values of $f(k)$.