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 ($\mathrm{\exists }n\in \mathrm{N}$, $1\le k\le n$) in Certain Intervals

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 ($where$ $\mathrm{\exists }n\in \mathrm{N}$, $1\le k\le n$) 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.