Number of ordered factorizations and recursive divisors

In this note we describe a new method of counting the number of unordered factorizations of a natural number by means of a generating function and a recurrence relation arising from it, which improves an earlier result in this direction.

We recently introduced the recursive divisor function $\kappa_x(n)$, a recursive analogue of the usual divisor function. Here we calculate its Dirichlet series, which is ${\zeta(s-x)}/(2 - \zeta(s))$. We show that $\kappa_x(n)$ is related to the ordinary divisor function by $\kappa_x * \sigma_y = \kappa_y * \sigma_x$, where * denotes the Dirichlet convolution. Using this, we derive several identities relating $\kappa_x$ and some standard arithmetic functions. We also clarify ...

We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: $\kappa_x(n) = n^x + \sum_{d\lfloor n} \kappa_x(d)$, where the sum is over the proper divisors of $n$. We give a geometrical interpretation of $\kappa_x(n)$, which we use to derive a relation between $\kappa_x(n)$ and $\kappa_0(n)$. For $x \geq 2$, we observe that $\kappa_x(n)/n^x < 1/(2-\zeta(x))$. We show that, for $n \geq 2$, $\kappa_0(n)$ is twice the number of ordered...

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...

As a well-known enumerative problem, the number of solutions of the equation $m=m_1+...+m_k$ with $m_1\leqslant...\leqslant m_k$ in positive integers is $\Pi(m,k)=\sum_{i=0}^k\Pi(m-k,i)$ and $\Pi$ is called the additive partition function. In this paper, we give a recursive formula for the so-called multiplicative partition function $\mu_1(m,k):=$ the number of solutions of the equation $m=m_1... m_k$ with $m_1\leqslant...\leqslant m_k$ in positive integers. In particular, us...

The $j$th divisor function $d_j$, which counts the ordered factorisations of a positive integer into $j$ positive integer factors, is a very well-known arithmetic function; in particular, $d_2(n)$ gives the number of divisors of $n$. However, the $j$th non-trivial divisor function $c_j$, which counts the ordered proper factorisations of a positive integer into $j$ factors, each of which is greater than or equal to 2, is rather less well-studied. We also consider associated di...

The partitions of the integers can be expressed exactly in an iterative and closed-form expression. This equation is derived from distributing the partitions of a number in a network that locates each partition in a unique and orderly position. From this representation an iterative equation for the function of the number of divisors was derivated. Also, the number of divisors of a integer can be found from a new function called the trace of the number n, trace(n). As a final ...

We study the number of factorizations of a positive integer, where the parts of the factorization are of l different colors (or kinds). Recursive or explicit formulas are derived for the case of unordered and ordered, distinct and non-distinct factorizations with at most and exactly l colors.

The details for the construction of an explicit formula for the divisors function d(n) = #{d | n} are formalized in this article. This formula facilitates a unified approach to the investigation of the error terms of the divisor problem and circle problem.

Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of representing $n$ as an ordered product of $j+r$ factors, of which the first $j$ must be non-trivial, and their natural extension to negative integers $r.$ We give recurrence properties and explicit formulae for these novel arithmetic functions. Speci...