Sequences, Series and Uniform distribution of SP Numbers

TL;DR

This paper investigates the properties and distribution patterns of SP numbers, which are numbers of the form p*a^2 with p prime and a > 1, including their sequences and related fractional sequences.

Contribution

It introduces new sequences related to SP numbers and analyzes their distribution and uniformity properties within the interval [0,1].

Findings

Sequences of SP numbers exhibit specific distribution patterns.

Some related fractional sequences are shown to be uniformly distributed.

The paper provides new insights into the distribution of these special numbers.

Abstract

We defined numbers of the form $p\cdot a^2$ as SP numbers (Square-Prime numbers) ($a\neq1$, $p$ prime) in 'Distribution of Square-Prime Numbers' (arXiv:2109.10238). These numbers are listed in the OEIS as A228056. Some examples of SP numbers: $75=3 \cdot 25; 108=3 \cdot 36; 45= 5 \cdot 9$. This paper explores sequences of these numbers, sequences between 0 and 1 related to these numbers and analyzes the distribution of some of these sequences.