Related papers: Pentagonal number recurrence relations for $p(n)$

Euler totient function $\phi(n)$ plays a central role in number theory and is applied in areas such as cryptography. In this paper, we study iterations of the totient function. We first prove that for any integer $n>2$, iteratively applying…

In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of 2 for the function $p_8(n)$ by finding…

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…

We study $\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\nu_2$ and $\nu_3$ take on values divisible by 2 and 4. Expanding earlier work, we show $\nu_2(An+B) \equiv 0…

In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer $n$ with even crank and those with odd crank, denoted $C(n)=c_e(n)-c_o(n)$. Inspired by Ramanujan's classical…

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…

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…

Suppose that $\theta$ is irrational. Then almost all elements $\nu\in {\mathbb Z}[\theta]$ that may be written as a $k$-fold product of the shifted integers $n+\theta$ $(n\in {\mathbb N})$ are thus represented essentially uniquely.

We prove new formulas and congruences for $p(n,k):=$ the number of partitions of $n$ into $k$ parts and $q(n,k):=$ the number of partitions of $n$ into $k$ distinct parts. Also, we give lower and upper bounds for the density of the set…

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Hirschhorn and Sellers showed that $\overline{p}(4n+3)\equiv 0 \pmod{8}$ for $n\geq 0$. They also conjectured that $\overline{p}(40n+35)\equiv 0 \pmod{40}$ for $n\geq 0$.…

A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result,…

For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities…

Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition…

Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{\deg f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by…

We prove that the partition function $p(n)$ is log-concave for all $n>25$. We then extend the results to resolve two related conjectures by Chen. The proofs are based on Lehmer's estimates on the remainders of the Hardy--Ramanujan and the…

Let $m$, $r$ and $n$ be positive integers. We denote by ${\bf k}\vdash n$ any tuple of odd positive integers ${\bf k}=(k_1,\dots,k_t)$ such that $k_1+\dots+k_t=n$ and $k_j\ge 3$ for all $j$. In this paper we prove that for every…

Let $T_\ell(n)$ denote the number of $\ell-$regular partition triples of $n$ and let $p_{\ell, 3}(n)$ enumerates the number of 2--color partition triples of $n$ where one of the colors appear only in parts that are multiples of $\ell$. In…

We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$)…

Let $\{B_n\}$, $\{B_n(x)\}$ and $\{E_n(x)\}$ be the Bernoulli numbers, Bernoulli polynomials and Euler polynomials, respectively. In this paper we mainly establish formulas for $\sum_{6\mid k-3}\binom nkB_{n-k}(x)$, $\sum_{6\mid k}\binom…

The enumeration $d_k(n)$ of $k$-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function $p(n)$. Congruences for $d_k(n)$ modulo certain powers of primes have been proven via elementary…