A note on the $\Sopfr(n)$ function

TL;DR

This paper explores the properties of the sum of prime factors function, $ ext{Sopfr}(n)$, drawing analogies with prime distribution and proposing conjectures related to prime numbers based on numerical analysis.

Contribution

It introduces a numerical study of $ ext{Sopfr}(n)$, highlighting its analogy with prime distribution and suggesting new conjectures about primes.

Findings

Numerical analysis of $ ext{Sopfr}(n)$ reveals patterns similar to prime distribution.

Proposes conjectures linking $ ext{Sopfr}(n)$ behavior to prime number properties.

Draws analogies between $ ext{Sopfr}(n)$ and prime counting functions.

Abstract

The $\Sopfr(n)$ function is defined as the sum of prime factors of $n$ each of which is taken with its multiplicity. This function is studied numerically. The analogy between $\Sopfr(n)$ and the primes distribution function is drawn and some conjectures for prime numbers formulated in terms of the $\Sopfr(n)$ function are suggested.