相关论文: Infinite sets of $b$-additive and $b$-multiplicati…

Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{\lambda-1}, b^{\lambda}-1\right]$, we denote by $R_\lambda (n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We…

For an integer $b\geq 2$, a positive integer is called a $b$-Niven number if it is a multiple of the sum of the digits in its base-$b$ representation. In this article, we show that every arithmetic progression contains infinitely many…

We derive the asymptotic formula for $p_n(N,M)$, the number of partitions of integer $n$ with part size at most $N$ and length at most $M$. We consider both $N$ and $M$ are comparable to $\sqrt{n}$. This is an extension of the classical…

The null-function $0(a):=0$, $\forall a\in $N, has Ramanujan expansions: $0(a)=\sum_{q=1}^{\infty}(1/q)c_q(a)$ (where $c_q(a):=$ Ramanujan sum), given by Ramanujan, and $0(a)=\sum_{q=1}^{\infty}(1/\varphi(q))c_q(a)$, given by Hardy…

Generalized taxicab numbers are the smallest positive integers that are the sum of exactly $j$, positive $k$-th powers in exactly $m$ distinct ways. This paper is considers for which values of $m$ does a smallest such integer exist as $j$…

The well-known Hardy--Ramanujan inequality states that if $\omega(n)$ denotes the number of distinct prime factors of a positive integer $n$, then there is an absolute constant $C>0$ such that uniformly for $x\ge2$ and $k\in\mathbb{N}$,…

The Hardy-Ramanujan formula for the number of integer partitions of $n$ is one of the most popular results in partition theory. While the unabridged final formula has been celebrated as reflecting the genius of its authors, it has become…

In this paper, we focus on Narayana numbers which can be written as a products of four repdigits in base $g$, where $g$ is an integer with $g\geq2$. We prove that for $g$ between $2$ and $12$, there are finitely many of these numbers.…

For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. In this paper, for a sequence of integers $B=\{1<b_1<b_2<\cdots\}$ and $3b_1+5\leq b_2\leq 6b_1+10$, we determine the critical value for $b_3$ such that there exists an…

Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results…

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…

A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n<=x that are in S, for…

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. For all integers $a$ and $b \geq 1$ with $\gcd(a, b) = 1$, let $[a^{-1} \!\bmod b]$ be the multiplicative inverse of $a$ modulo $b$, which we pick in the usual set of…

In this paper, we consider some additive properties of integers with restricted digit expansions. Let $b\geq 3$ be an integer and $B_b$ be the set of integers whose base $b$ expansions have only digits $\{0,1\}.$ Let $a,b,c$ be three…

Let $b$ be a numeration base. A $b$-Niven number is one that is divisible by the sum of its base $b$ digits. We introduce high degree $b$-Niven numbers. These are $b$-Niven numbers that have a power greater than $1$ that is $b$-Niven…

In this paper, we prove that for any $a,M\in \mathbb N$ with $(a,M)=1$, there are infinitely many Carmichael numbers $m$ such that $m\equiv a$ mod $M$

We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and…

An MSTD set is a finite set of integers with more sums than differences. It is proved that, for infinitely many positive integers $k$, there are infinitely many affinely inequivalent MSTD sets of cardinality $k$. There are several related…

In DOI:10.1017/etds.2022.2 the author proved that for each integer $k$ there is an implicit number $M > 0$ such that if $b_1, \cdots , b_k$ are multiplicatively independent integers greater than $M$, there are infinitely many integers whose…

In this paper we prove that if $A$ and $B$ are infinite subsets of positive integers such that every positive integer $n$ can be written as $n=ab$, $a\in A$, $b\in B$, then $\displaystyle \lim_{x\to \infty}\frac{A(x)B(x)}{x}=\infty $. We…