On a Deterministic Property of the Category of $k$-almost Primes: A Deterministic Structure Based on a Linear Function for Redefining the $k$-almost Primes ($\exists n\in {\rm N} $, $1{\le} k {\le}n$) in Certain Intervals

TL;DR

This paper introduces a linear-function-based deterministic algorithm for identifying all $k$-almost primes within specific intervals, revealing a new algebraic property and relations that enhance understanding of these numbers.

Contribution

It presents a novel deterministic algorithm and algebraic characterization for $k$-almost primes, offering new insights and relations in their structure.

Findings

Proven a new deterministic property of $k$-almost primes.

Developed a simple algebraic algorithm for $k$-almost primes in intervals.

Discovered relations containing new information about $k$-almost primes.

Abstract

In this paper based on a sort of linear function, a deterministic and simple algorithm with an algebraic structure is presented for calculating all (and only) $k$-almost primes ($where$ $\exists n\in {\rm N} $, $1{\le} k {\le}n$) in certain interval. A theorem has been proven showing a new deterministic property of the category of $k$-almost primes. Through a linear function that we obtain, an equivalent redefinition of the $k$-almost primes with an algebraic characteristic is identified. Moreover, as an outcome of our function's property some relations which contain new information about the $k$-almost primes (including primes) are presented.