Algebraic Results on SP Numbers along with a generalization

TL;DR

This paper explores algebraic properties of SP numbers, which are numbers of the form p*a^2 with p prime, and generalizes their definition to include arbitrary natural number powers, providing new theoretical insights.

Contribution

It introduces algebraic theorems about SP numbers and extends their definition to broader classes involving arbitrary natural powers.

Findings

Proved algebraic properties of SP numbers

Generalized the definition to include arbitrary powers

Connected SP numbers to existing OEIS sequence

Abstract

We defined numbers of the form $p\cdot a^2$ as SP numbers (Square-Prime numbers) ($a\neq1$, $p$ prime) in the paper 'Distribution of Square-Prime numbers' (arXiv:2109.10238) along with proofs on their distribution. Some examples of SP Numbers : 75 = 3 $\cdot$ 25; 108 = 3 $\cdot$ 36; 45 = 5 $\cdot$ 9. These numbers are listed in the OEIS as A228056. In this paper, we will prove a few algebraic theorems and generalize the definition of SP Numbers to allow factors of arbitrary natural number powers.