Number of ordered factorizations and recursive divisors

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset $S$ of $\mathbb{Z}$. Bhargava's factorials $k!_S$ are invariants, constructed using the notion of $p$-orderings of $S$ where $p$ is a prime. This paper defines $b$-orderings of any nonempty subset $S$ of $\mathbb{Z}$ for all integers $b\ge2$; the integers $b\ge2$ correspond to the nonzero proper ideals in $\mathbb{Z}$. This paper defines generalized factorials $k !_{S,T}$ and...

Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function $\sigma(n)$. In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a calculus to associate a generating function with each of these divisor sums. This yields analogues of Ramanujan's recurrence relation for several partition-theoretic functions as well as $r_k(n)$ and $t_k(n)$, functions counting the number ...

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results of some special values of the sums of divisors functions.

We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) som...

We establish formulas for the constant factor in several asymptotic estimates related to the distribution of integer and polynomial divisors. The formulas are then used to approximate these factors numerically.

We prove new variants of the Lambert series factorization theorems studied by Merca and Schmidt (2017) which correspond to a more general class of Lambert series expansions of the form $L_a(\alpha, \beta, q) := \sum_{n \geq 1} a_n q^{\alpha n-\beta} / (1-q^{\alpha n-\beta})$ for integers $\alpha, \beta$ defined such that $\alpha \geq 1$ and $0 \leq \beta < \alpha$. Applications of the new results in the article are given to restricted divisor sums over several classical speci...

We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.

Let $f(n)$ denote the number of distinct unordered factorisations of the natural number $n$ into factors larger than 1.In this paper, we address some aspects of the function $f(n)$.

We prove new exact formulas for the generalized sum-of-divisors functions, $\sigma_{\alpha}(x) := \sum_{d|x} d^{\alpha}$. The formulas for $\sigma_{\alpha}(x)$ when $\alpha \in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over all of the prime factors $n \leq x$ and terms involving the $r$-order harmonic number sequences and the Ramanujan sums $c_d(x)$. The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when $r >...