Ordered Factorizations with $k$ Factors

TL;DR

This paper explores the combinatorial properties of ordered k-factorizations of integers, revealing polynomial behaviors in their cumulative sums and deriving average orders based on divisor problem results.

Contribution

It provides explicit polynomial expressions for the cumulative sum of ordered factorizations and connects these to divisor problem results for fixed k.

Findings

Cumulative sum of factorizations is a polynomial in log base l of x.

Explicit formulas for polynomial degree and coefficients are given.

Average order derived from divisor problem results.

Abstract

We give an overview of combinatoric properties of the number of ordered $k$-factorizations $f_k(n,l)$ of an integer, where every factor is greater or equal to $l$. We show that for a large number $k$ of factors, the value of the cumulative sum $F_k(x,l)=\sum\nolimits_{n\leq x} f_k(n,l)$ is a polynomial in $\lfloor \log_l x \rfloor$ and give explicit expressions for the degree and the coefficients of this polynomial. An average order of the number of ordered factorizations for a fixed number $k$ of factors greater or equal to 2 is derived from known results of the divisor problem.