Related papers: Partition functions that repel perfect-powers

In 2013 Zhi-Wei Sun conjectured that $p(n)$ is never a power of an integer when $n>1.$ We confirm this claim in many cases. We also observe that integral powers appear to repel the partition numbers. If $k>1$ and $\Delta_k(n)$ is the…

In 2013, Sun conjectured that the partition function $p(n)$ is never a perfect power for $n \geq 2$. Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers $d \geq 0$ and $k \geq 2$, there appear to be only…

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number $\beta$. We prove that if $\beta$ is real quadratic, then the number of partitions is always finite if and only if some conjugate of $\beta$…

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 $a_1 + >... + a_k = n$…

The study of the well-known partition function $p(n)$ counting the number of solutions to $n = a_{1} + \dots + a_{\ell}$ with integers $1 \leq a_{1} \leq \dots \leq a_{\ell}$ has a long history in combinatorics. In this paper, we study a…

One of the most basic results concerning the number-theoretic properties of the partition function $p(n)$ is that $p(n)$ takes each value of parity infinitely often. This statement was first proved by Kolberg in 1959, and it was…

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…

Given integer $n > 0$ and $m > 1$, we call a partition of set $[n] = \{1, \dots, n\}$ {\em $m$-good} if each of the partitioning sets is of size at most $m$ and the sum of numbers in it is a power of $m$, that is, $m^t$ for some $t \geq 0$.…

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the…

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…

The number partition problem is a well-known problem, which is one of 21 Karp's NP-complete problems \cite{karp}. The partition function is a boolean function that is equivalent to the number partition problem with number range restricted.…

The aim of this note is to provoke discussion concerning arithmetic properties of function $p_{d}(n)$ counting partitions of an positive integer $n$ into $d$-th powers, where $d\geq 2$. Besides results concerning the asymptotic behavior of…

Extending the partition function multiplicatively to a function on partitions, we show that it has a unique maximum at an explicitly given partition for any $n\neq 7$. The basis for this is an inequality for the partition function which…

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$ p(m^j\ell^kn+B)\equiv 0\mod m \text{for all} n\ge 0 $$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell\neq 2,3,m$.…

For each $n\geq 1$, we express the partition function $p(n)$ as a CM trace on $X_0(6)$ of the discriminant $\Delta_n:=1-24n$ invariants of a weight 0 weak Maass function $P$ that records where CM elliptic curves sit on $X(1)$, together with…

The main result of the paper is the Fibonacci-like property of the partition function. The partition function $p(n)$ has a property: $p(n) \leq p(n-1) + p(n-2)$. Our result shows that if we impose certain restrictions on the partition, then…

For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…

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…

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…

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…