Distribution of Square-Prime Numbers

Abstract

For $a \neq 1$ and $p$ a 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.