Number of ordered factorizations and recursive divisors

TL;DR

This paper derives explicit closed-form expressions for the number of ordered factorizations and recursive divisors, revealing their structure and properties through geometric and hypergeometric function representations.

Contribution

It introduces three closed-form formulas for these recursive arithmetic functions, connecting them to generalized hypergeometric functions and providing new insights.

Findings

Explicit formulas for ordered factorizations and recursive divisors

Connection to generalized hypergeometric functions

Enhanced understanding of their number-theoretic properties

Abstract

The number of ordered factorizations and the number of recursive divisors are two related arithmetic functions that are recursively defined. But it is hard to construct explicit representations of these functions. Taking advantage of their recursive definition and a geometric interpretation, we derive three closed-form expressions for them both. These expressions shed light on the structure of these functions and their number-theoretic properties. Surprisingly, both functions can be expressed as simple generalized hypergeometric functions.