Related papers: A note on congruences for generalized cubic partit…

For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form…

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 any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2…

Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such…

The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these congruence results apply to the generalized Frobenius partitions…

Ramanujan proved three famous congruences for the partition function modulo 5, 7, and 11. The first author and Boylan proved that these congruences are the only ones of this type. In 1984 Andrews introduced the $m$-colored Frobenius…

We introduce two new integer partition functions, both of which are the number of partition quadruples of $n$ with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$ by use…

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\phi_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved…

For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5\lambda+1),…

Let $c\phi_{k}(n)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $c\phi_{3}(n)$ and $c\phi_{9}(n)$ modulo arbitrary powers of 3, which refine the results of…

The partition function $ p_{[1^c\ell^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{\ell}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{\ell n})^d}.\] In \cite{P}, we proved infinite family of…

In this short note, we prove several new congruences for the overcubic partition triples function, using both elementary techniques and the theory of modular forms. These extend the recent list of such congruences given by Nayaka,…

Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form $p( Q^3 \ell n+\beta)\equiv0\pmod\ell$ where $\ell$ and $Q$ are prime and $5\leq \ell\leq 31$; these lie in two natural…

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4…

We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes.…

Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of…

The notion of cubic partitions is introduced by Hei-Chi Chan and named by Byungchan Kim in connection with Ramanujan's cubic continued fractions. Chan proved that cubic partition function has Ramanujan Type congruences modulo powers of $3$.…

The main result of the paper is the existence of an infinitely many families of Ramanujan-type congruences for $b_4(n)$ and $b_6(n)$ modulo primes $m \geq 2$ and $m \geq 5$, respectively. We provide new examples of congruences for $b_4(n)$…

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\phi_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved…

Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+\beta) \equiv 0 \pmod{\ell}.$…