AbstractLet p1,p2,…,pn be distinct primes. In 1970, Erdős, Herzog and Schönheim proved that if D, |D|=m, is a set of divisors of N=p1α1⋯pnαn, α1≥α2≥⋯≥αn, no two members of the set being coprime and if no additional member may be included in D without contradicting this requirement then m≥αn∏i=1n−1(αi+1). They asked to determine all sets D such that the equality holds. In this paper we solve this problem. We also pose several open problems for further research
We define the arithmetic function P by P (1) = 0, and P (n) = p1 + p2+ · · ·+ pk if n has the uni...
AbstractWe find the formula for the cardinality of a maximal set of integers from {1,…,n} which does...
AbstractLet m,n1,n2,…,nm and c be positive integers. Let A = {A1, A2, … Am} be a system of sequences...
AbstractMain result: Let ƒ be a collection of divisors of N = pe11⋯penn (e1 = min ei for all i ϵ {1,...
AbstractLet A={a1, a2, …}⊆N and put A(n)=∑ai⩽n1. We say that A is a P-set if no element ai divides t...
Abstract. An amicable pair consists of two distinct numbers N and M each of which is the sum of the ...
AbstractIt is shown that if F = [D1, D2,…, Ds] is a set of divisors of a number N such that each div...
If $\mathcal{A}\subset\mathbb{N}$ is such that it does not contain a subset $S$ consisting of $k$ pa...
We prove that for all odd primes $p$ and positive integers $\alpha \geq 2$, a construction of Batte...
Ahlswede R, Blinovsky V. Maximal sets of numbers not containing k+1 pairwise coprimes and having div...
Abstract. For positive integers n and k, it is possible to choose primes P1, P2, · · · , Pk such ...
Given an integer n ≥ 3, let u1, …, un be pairwise coprime integers ≥ 2, D a family of nonempty prope...
AbstractErdős estimated the maximal number of integers selected from {1,2,…,N}, so that none of them...
AbstractThe expressions ϕ(n)+σ(n)−3n and ϕ(n)+σ(n)−4n are unusual among linear combinations of arith...
AbstractThe purpose of this note is to show that, if n1,…, nkare positive integers, and for each d ϵ...
We define the arithmetic function P by P (1) = 0, and P (n) = p1 + p2+ · · ·+ pk if n has the uni...
AbstractWe find the formula for the cardinality of a maximal set of integers from {1,…,n} which does...
AbstractLet m,n1,n2,…,nm and c be positive integers. Let A = {A1, A2, … Am} be a system of sequences...
AbstractMain result: Let ƒ be a collection of divisors of N = pe11⋯penn (e1 = min ei for all i ϵ {1,...
AbstractLet A={a1, a2, …}⊆N and put A(n)=∑ai⩽n1. We say that A is a P-set if no element ai divides t...
Abstract. An amicable pair consists of two distinct numbers N and M each of which is the sum of the ...
AbstractIt is shown that if F = [D1, D2,…, Ds] is a set of divisors of a number N such that each div...
If $\mathcal{A}\subset\mathbb{N}$ is such that it does not contain a subset $S$ consisting of $k$ pa...
We prove that for all odd primes $p$ and positive integers $\alpha \geq 2$, a construction of Batte...
Ahlswede R, Blinovsky V. Maximal sets of numbers not containing k+1 pairwise coprimes and having div...
Abstract. For positive integers n and k, it is possible to choose primes P1, P2, · · · , Pk such ...
Given an integer n ≥ 3, let u1, …, un be pairwise coprime integers ≥ 2, D a family of nonempty prope...
AbstractErdős estimated the maximal number of integers selected from {1,2,…,N}, so that none of them...
AbstractThe expressions ϕ(n)+σ(n)−3n and ϕ(n)+σ(n)−4n are unusual among linear combinations of arith...
AbstractThe purpose of this note is to show that, if n1,…, nkare positive integers, and for each d ϵ...
We define the arithmetic function P by P (1) = 0, and P (n) = p1 + p2+ · · ·+ pk if n has the uni...
AbstractWe find the formula for the cardinality of a maximal set of integers from {1,…,n} which does...
AbstractLet m,n1,n2,…,nm and c be positive integers. Let A = {A1, A2, … Am} be a system of sequences...
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