Related papers: Divisibility tests with weighted digital sums

We study the connection between the Mersenne numbers $M(n) = 2^n-1$ and the dynamics of the angle-doubling map. Within this framework, we develop an algorithm to compute divisors of Mersenne numbers without explicitly evaluating $M(n)$.…

In this paper, we use the transference principle to investigate the representation of sufficiently large positive integers as the sum of prime powers and integer powers, where the primes are drawn from a positive density subset of the set…

We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a…

Odd numbers can be indexed by the map k(n)=(n-3)/2, n belonging to 2N+3. We first propose a basic primality test using this index function that was first introduced in article (8). Input size of operations is reduced which improves…

Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides…

We study the average value of the divisor function $\tau(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$.…

Let $k\ge 2$ and $\Pi(n)=\prod_{i=1}^k(a_in+b_i)$ for some integers $a_i, b_i$ ($1\le i\le k$). Suppose that $\Pi(n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that $\Omega(\Pi(n))\le r_k$…

Let $n$ and $k$ be positive integers and $\sigma(n)$ the sum of all positive divisors of $n$. We call $n$ an exactly $k$-deficient-perfect number with deficient divisors $d_1, d_2, \ldots, d_k$ if $d_1, d_2, \ldots, d_k$ are distinct proper…

We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from $s$ distributions, $p_1, p_2, \ldots, p_s$, we design testers for the…

The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…

We investigate three combinatorial problems considered by Erd\"os, Rivat, Sark\"ozy and Sch\"on regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this…

This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…

Let $X_1,X_2,...$ be the digits in the base-$q$ expansion of a random variable $X$ defined on $[0,1)$ where $q\ge2$ is an integer. For $n=1,2,...$, we study the probability distribution $P_n$ of the (scaled) remainder…

A positive integer $n$ is said to be $k$-layered if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we start the systematic study of these class of numbers. In particular, we state some algorithms to find some…

We give a separability criterion for three qubit states in terms of diagonal and anti-diagonal entries. This gives us a complete characterization of separability when all the entries are zero except for diagonal and anti-diagonals. The…

For given integers a,b, and j at least 1 we determine the set of integers n for which a^n-b^n is divisible by n^j. For j=1,2, this set is usually infinite; we find explicitly the exceptional cases for which a,b the set is finite. For j=2,…

We use sums of Liouville type to count the number of ways a positive integer can be represented by the forms $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \bigl(k(b+d) \bigr)^{1/3} y$, and $\bigl(k(a+c) \bigr)^{1/3} x + l(b+d) y$ for nonnegative…

Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values…

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 1/2, then every sufficiently large even integer can be written as the sum of eight primes from A. The constant 1/2 in this statement is…

In this paper we propose necessary upper bounds for the second, third and fourth smallest prime divisors of friends of 10 based on the number of distinct prime divisors of it.