On the counts of p-rough numbers

TL;DR

This paper investigates the properties of p-rough numbers, revealing a symmetry in their counting function and introducing a new periodic, bounded difference function with rotational symmetry.

Contribution

It demonstrates a symmetry in the counting function of p-rough numbers and defines a new related function with notable periodic and symmetric properties.

Findings

Existence of a symmetry line in the function (x,p)

Introduction of the (x,p) difference function with periodicity

The difference function exhibits rotational symmetry

Abstract

The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line of symmetry for the function $\Phi(x,p)$, and we introduce the function $\Delta \Phi(x,p)$ which is the difference between $\Phi(x,p)$ and the line of symmetry. $\Delta \Phi(x,p)$ is periodic and bounded and has a rotational symmetry.