Distribution of Square-Prime Numbers

Abstract

For a 6= 1 and p prime, we define numbers of the form pa to be Square-Prime (SP) Numbers. For example, 75 = 3 · 25; 108 = 3 · 36; 45 = 5 · 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.