On $A{P}_3$ - covering sequences

Abstract

Recently, motivated by Stanley sequences, Kiss, S\' andor and Yang introduced a new type sequence: a sequence $A$ of nonnegative integers is called an $A{P}_k$ - covering sequence if there exists an integer $n_0$ such that if $n>n_0$, then there exist $a_1\in A,\dots ,a_{k-1}\in A$, $a_1<a_2<\cdots <a_{k-1}<n$ such that $a_1,\dots ,a_{k-1},n$ form a $k$-term arithmetic progression. They prove that there exists an $A{P}_3$ - covering sequence $A$ such that $\underset{n\to \infty }{\mathrm{limsup}}A(n)/\sqrt{n}\le 34$. In this note, we prove that there exists an $A{P}_3$ - covering sequence $A$ such that $\underset{n\to \infty }{\mathrm{limsup}}A(n)/\sqrt{n}=\sqrt{15}$.

5 pages

References in corpus (1)

• On the Growth of the Counting Function of Stanley Sequences