Distribution of Square Prime Numbers

TL;DR

This paper investigates the properties and distribution of Square-Prime numbers, defined as numbers of the form p times a squared integer, analyzing their density and verifying conjectures through computational methods.

Contribution

It introduces the concept of Square-Prime numbers, studies their distribution and density, and verifies related conjectures using computational tools.

Findings

Distribution patterns of Square-Prime numbers analyzed

Density estimates of these numbers provided

Conjectures verified up to large bounds using computer programs

Abstract

For $a \neq 1$ and $p$ prime, we define numbers of the form $pa^2$ to be Square-Prime (SP) Numbers. For example, 75 = 3 $\cdot$ 25; 108 = 3 $\cdot$ 36; 45 = 5 $\cdot$ 9. These numbers are listed in the OEIS as A228056. We study the properties of these numbers, their distribution/density and also develop a few claims on their distribution/density. We rely on computer programs to verify some conjectures up to large numbers.