Related papers: Parity of 4-regular and 8-regular partition functi…

Let $f(n)$ be the number of distinct exponents in the prime factorization of the natural number $n$. We prove some results about the distribution of $f(n)$. In particular, for any positive integer $k$, we obtain that $$ \#\{n \leq x : f(n)…

Recently, the concept of parity bias in integer partitions has been studied by several authors. We continue this study here, but for non-unitary partitions (namely, partitions with parts greater than $1$). We prove analogous results for…

Jing Run Chen proved in 1966 that $p+2$ has at most two prime factors for infinitely many primes $p$. However, due to the parity problem we do not know whether $p+2$ has an odd (or even) number of prime factors infinitely often. In the…

Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the property that corresponding to every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results…

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.

We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$ such that $\pi_1$ and $\pi_2$ comprise distinct odd parts and…

A Goellnitz-Gordon partition is one in which the parts differ by at least 2, and where the inequality is strict if a part is even. Let Q_i(n) denote the number of partitions of n into distinct parts not congruent to i mod 4. By attaching…

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…

In his classic text, \emph{Combinatory Analysis}, MacMahon defined a perfect partition of a positive integer $n$ as a partition whose parts contain exactly one partition of every positive integer not exceeding $n$. In this paper we apply…

Let \pi be a partition. In [2] we defined BG-rank(\pi) as an alternating sum of parities of parts. This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let p_j(n)(a_{t,j}(n)) denote a…

Recently, several mathematicians have investigated various partition functions with the goal of discovering Ramanujan-type congruences. One such function is $\overline{B}_{2^\alpha}(n)$, which represents the number of $2^\alpha-$regular…

An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect…

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find…

Let f in Z[x,y] be a reducible homoegeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.

An odd prime $p$ is called irregular with respect to Euler polynomials if it divides the numerator of one of the numbers $$E_1(0),E_{3}(0),\ldots,E_{p-2}(0),$$ where $E_n(x)$ is the $n$-th Euler polynomial. As in the classical case, we link…

Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…

In 2017, Keith presented a comprehensive survey on integer partitions into parts that are simultaneously regular, distinct, and/or flat. Recently, the authors initiated a study of partitions into parts that are simultaneously regular and…

Let $\mathfrak{p}_{\mathbb{P}_r}(n)$ denote the number of partitions of $n$ into $r$-full primes. We use the Hardy-Littlewood circle method to find the asymptotic of $\mathfrak{p}_{\mathbb{P}_r}(n)$ as $n \to \infty$. This extends previous…

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…