Related papers: On Euler's Theorem

For a function $f\colon \mathbb{N}\to\mathbb{N}$, define $N^{\times}_{f}(x)=\#\{n\leq x: n=kf(k) \mbox{ for some $k$} \}$. Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…

A classic theorem of Uchimura states that the difference between the sum of the smallest parts of the partitions of $n$ into an odd number of distinct parts and the corresponding sum for an even number of distinct parts is equal to the…

Franklin's identity generalizes Euler's identity and states that the number of partitions of $n$ with $j$ different parts divisible by $r$ equals the number of partitions of $n$ with $j$ repeated parts. In this article, we give a refinement…

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left( C \sqrt{\frac{\log…

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…

Inspired by Armin Straub's conjecture (arXiv:1601.07161) about the number and maximal size of (2n+1, 2n+3)-core partitions with distinct parts, we develop relatively efficient, symbolic-computational algorithms, based on non-linear…

Let A and M be nonempty sets of positive integers. A partition of the positive integer n with parts in A and multiplicities in M is a representation of n in the form n = \sum_{a\in A} m_a a, where m_a is in M U {0} for all a in A, and m_a…

We obtain a unification of two refinements of Euler's partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson partition identity is used…

For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\sum_{k=0}^{[n/2]}\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$…

Let $m\ge 2$ be a fixed positive integer. Suppose that $m^j \leq n< m^{j+1}$ is a positive integer for some $j\ge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this…

Define a "nuclear partition" to be an integer partition with no part equal to one. In this study we prove a simple formula to compute the partition function $p(n)$ by counting only the nuclear partitions of $n$, a vanishingly small subset…

This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger…

Let q be an odd positive integer and P \in F2[z] be of order q and such that P(0) = 1. We denote by A = A(P) the unique set of positive integers satisfying \sum_{n=0}^\infty p(A, n) z^n \equiv P(z) (mod 2), where p(A,n) is the number of…

Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal{A} \subseteq \mathcal{B}(n)$ is a non-trivial $t$-intersecting family of set partitions i.e. any two members of $\A$ have at least $t$ blocks in…

A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the…

We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts…

In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an…

In this note, we obtain a formula which leads to a practical and efficient method to calculate the number of partitions of n into parts not divisible by m for given natural numbers n and m.

For a set of nonnegative integers $S$ let $R_{S}(n)$ denote the number of unordered representations of the integer $n$ as the sum of two different terms from $S$. In this paper we focus on partitions of the natural numbers into two sets…

For $k\geq i\geq 1$, let $B_{k,i}(n)$ denote the number of partitions of $n$ such that part 1 appears at most $i-1$ times, two consecutive integers l and $l+1$ appear at most $k-1$ times and if l and $l+1$ appear exactly $k-1$ times then…