Related papers: On unit-weighted zero-sum constants of $\mathbb Z_…

The constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of consecutive terms whose $A$-weighted sum is zero, where the weight set $A\subseteq \mathbb…

For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset…

For a weight-set $A\subseteq \mathbb Z_n$, the $A$-weighted zero-sum constant $C_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence of…

For $A\subseteq \mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of length $n$, whose $A$-weighted sum is zero.…

For $A\subseteq\mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of length $n$ whose $A$-weighted sum is zero.…

Let $A\subseteq \mathbb Z_n$ be a subset. A sequence $S=(x_1,\ldots,x_k)$ in $\mathbb Z_n$ is said to be an $A$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ such that $a_1x_1+\cdots+a_kx_k=0$. By a square, we shall mean a…

Let $A,B\subseteq \mathbb Z_n\setminus\{0\}$. A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_n$ is called an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that…

Let $A,B\subseteq\mathbb Z_n$ be given and $S=(x_1,\ldots, x_k)$ be a sequence in $\mathbb Z_n$. We say that $S$ is an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that…

Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $\mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $\ell$ such that any sequence over $G$ of length at least $\ell$ has an…

Let $G$ be a finite abelian group of exponent $n$ and let $A$ be a non-empty subset of $[1,n-1]$. The Davenport constant of $G$ with weight $A$, denoted by $D_A(G)$, is defined to be the least positive integer $\ell$ such that any sequence…

Let $G$ be a group and $A\subseteq [1,\exp(G)-1]$. We define the constant ${\sf C}_A(G),$ which is the least positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has an $A$-weighted consecutive product-one…

A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_p$ is called a $(Q_p,\mathbf 1)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in Q_p$ such that $a_1x_1+\cdots+a_kx_k=0$ and $a_1+\cdots+a_k=0$. The constant $E_{Q_p,\mathbf 1}$ is…

Let $C_n$ be the cyclic group of order $n$ and set $s_{A}(C_n^r)$ as the smallest integer $\ell$ such that every sequence $\mathcal{S}$ in $C_n^r$ of length at least $\ell$ has an $A$-zero-sum subsequence of length equal to $\exp(C_n^r)$,…

Let $G$ be a finite additive abelian group with exponent $d^kn, d,n>1,$ and $k$ a positive integer. For $S$ a sequence over $G$ and $A=\{1,2,\ldots,d^kn-1\}\setminus\{d^kn/d^i:i\in[1,k]\}, $ we investigate the lower bound of the number…

Let $G$ be a finite group. A finite collection of elements from $G$, where the order is disregarded and repetitions are allowed, is said to be a product-one sequence if its elements can be ordered such that their product in $G$ equals the…

The Ulam sequence is defined as $a_1 =1, a_2 = 2$ and $a_n$ being the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives $$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47,…

The Ulam sequence is given by $a_1 =1, a_2 = 2$, and then, for $n \geq 3$, the element $a_n$ is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence $1, 2,…

The classical Ulam sequence is defined recursively as follows: $a_1=1$, $a_2=2$, and $a_n$, for $n > 2$, is the smallest integer not already in the sequence that can be written uniquely as the sum of two distinct earlier terms. This…

Let $C_n$ be a cyclic group of order $n$. A sequence $S$ of length $\ell$ over $C_n$ is a sequence $S = a_1\boldsymbol\cdot a_2\boldsymbol\cdot \ldots\boldsymbol\cdot a_{\ell}$ of $\ell$ elements in $C_n$, where a repetition of elements is…

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erd\H{o}s states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known…