Related papers: Some New Congruences For Overpartition Function Wi…

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 $\ell\neq 2,3,m$.…

Let $S_l(M,N)$ denote a set of $\ell$ triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between a subset of $S_l(M,N)$ with (integral) parametric elements and…

Let ${{\overline{p}}_{3}}(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\overline{p}}_{3}(n)$ modulo small powers of 2, such as…

A partition of a positive integer $n$ is a non-increasing sequence of positive integers which sum to $n$. A recently studied aspect of partitions is the minimal excludant of a partition, which is defined to be the smallest positive integer…

For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo…

We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for…

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove…

In the 1960s Atkin discovered congruences modulo primes $\ell\leq 31$ for the partition function $p(n)$ in arithmetic progressions modulo $\ell Q^3$, where $Q\neq \ell$ is prime. Recent work of the first author with Allen and Tang shows…

The starting point for this work is the family of functions $\overline{p}_{-t}(n)$ which counts the number of $t$--colored overpartitions of $n.$ In recent years, several infinite families of congruences satisfied by $\overline{p}_{-t}(n)$…

Let $\mathrm{pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\ge…

Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer…

Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands…

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…

We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…

It is known that the Scholz conjecture on addition chains is true for all integers $n$ with $\ell(2n) = \ell(n)+1$. There exists infinitely many integers with $\ell(2n) \leq \ell(n)$ and we don't know if the conjecture still holds for them.…

In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the…

We prove several congruences satisfied by the generalized cubic and generalized overcubic partition functions, recently introduced by Amdeberhan, Sellers, and Singh. We also prove infinite families of congruences modulo powers of $2$ and…

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…

Inspired by Andrews' and Bachraoui's work on partitions with repeated smallest part, we extend the concept to overpartitions. We study overpartitions with the restriction that the smallest non-overlined part appears exactly $k$ times and…

For a given prime $p$, we study the properties of the $p$-dissection identities of Ramanujan's theta functions $\psi(q)$ and $f(-q)$, respectively. Then as applications, we find many infinite family of congruences modulo 2 for some…