A note on $$AP_3$$-covering sequences

Abstract

For a given integer $$k\ge 3$$, a sequence A of nonnegative integers is called an $$AP_k$$-covering sequence if there exists an integer $$n_0$$ such that, if $$n>n_0$$, then there exist $$a_1\in A$$, $$\ldots $$, $$a_{k-1}\in A$$, $$a_1<a_2<\cdots<a_{k-1}<n$$ such that $$a_1,a_2,\ldots ,a_{k-1},n$$ form a k-term arithmetic progression. For $$k=3$$, Kiss, Sandor and Yang observed that $$\limsup _{n\rightarrow \infty } A(n)/\sqrt{n}\ge 1.77$$ hold for any $$AP_3$$-covering sequence A. They also proved that there exists an $$AP_3$$-covering sequence A such that $$\limsup _{n\rightarrow \infty } A(n)/\sqrt{n}\le 36$$. Recently, Chen proved that there exists an $$AP_3$$-covering sequence A such that $$\limsup _{n\rightarrow \infty } A(n)/\sqrt{n}=\sqrt{15}=3.87\ldots $$. In this note, we prove that there exists an $$AP_3$$-covering sequence A such that $$\limsup _{n\rightarrow \infty } A(n)/\sqrt{n}=8/\sqrt{5}=3.57\ldots $$.