相关论文: A formula for the $r$-coloured partition function …

In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For…

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$…

Let $$\sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}$$ denote the sum of inverses of odd divisors of a positive integer $n$, and let $c_{r}(n)$ be the number of representations of $n$ as a sum of $r$ squares where representations with…

For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0,…

For $r \in \mathbb{R}, r> 1$ and $n \in \mathbb{Z}^+$, the divisor function $\sigma_{-r}$ is defined by $\sigma_{-r}(n) := \sum_{d \vert n} d^{-r}$. In this paper we show the number $C_r$ of connected components of…

Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \sum_{n\le x} \tau(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}), $$ for any integer $r\ge 2$. Here, $$ C_{r}=\frac{1}{(2^r-1)!}…

The divisor function $\sigma(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for…

Let $\sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$…

The partition function, $p_A(n)$, is defined to be the number of partitions of $n$ with parts in the set A, where $n$ is a positive integer and $A$ is a set of positive integers. It is well documented that: if A is a finite set with…

For any real number $s$, let $\sigma_s$ be the generalized divisor function, i.e., the arithmetic function defined by $\sigma_s(n) := \sum_{d \, \mid \, n} d^s$, for all positive integers $n$. We prove that for any $r > 1$ the topological…

Recently, Merca and Schmidt found some decompositions for the partition function $p(n)$ in terms of the classical M\"{o}bius function as well as Euler's totient. In this paper, we define a counting function $T_k^r(m)$ on the set of…

We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of numbers, illustrate how different mathematical fields play…

One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which…

Let $p(n)$ denote the partition function. DeSalvo and Pak proved that $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)> \frac{p(n)}{p(n+1)}$ for $n\geq 2$, as conjectured by Chen. Moreover, they conjectured that a sharper inequality…

Let $\mathbf a=(a_1,\ldots,a_r)$ be a vector of positive integers. In continuation of a previous paper we present other formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$…

For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details…

This note rederives a formula for $r$-color partitions, $1 \le r \le 24$, including Rademacher's celebrated result for ordinary partitions, from the duality between modular forms of weights $-r/2$ and $2+r/2$.

N. Minculete has introduced a concept of divisors of order $r$: integer $d=p_1^{b_1}\cdots p_k^{b_k} $ is called a divisor of order $r$ of $n=p_1^{a_1}\cdots p_k^{a_k}$ if $d \mid n$ and $b_j\in\{r, a_j\}$ for $j=1,\ldots,k$. One can…

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a…

Let $d_1 = 1 < d_2 < d_3 < \cdots < d_{\tau(n)} = n$ denote the increasing sequence of the divisors of a positive integer $n$. In this paper, for real or complex values of $\alpha$, we define and study some properties of two new divisor…