Let \({\mathbb {N}}\) be the set of positive integers and \(k\ge 3\) be an integer. Define the set \(G\subseteq (0,1]\) of real numbers as a k-good set if G contains no geometric progression of length k with some ratio \(r\in {\mathbb {N}}\setminus \{1\}\). The real number \(x\in (0,1]\setminus G\) is k-bad with respect to G if there exists an integer \(r\in {\mathbb {N}}\setminus \{1\}\) such that \(G\cup \{x\}\) contains the k-term geometric progression \((x,xr,xr^2,\ldots ,xr^{k-1})\). Define \(Bad(G)=\{x\in (0,1]\setminus G: x \quad \text{ is } \quad k\text{-bad } \text{ with } \text{ respect } \text{ to }\quad G\}\). In 2015, Nathanson and O’Bryant (Acta Arith 170:327–342, 2015) proved that there exists a unique strictly increasing sequence \(\big \{A_1^{(k)}<A_2^{(k)}<\cdots \big \}\) of positive integers with \(A_1^{(k)}=1\) such that \(G^{(k)}=\bigcup _{i=1}^{\infty } \big (1/ A_{2i}^{(k)},1/ A_{2i-1}^{(k)}\big ]\) is a k-good set and \(Bad(G^{(k)})=\bigcup _{i=1}^{\infty } \big (1/ A_{2i+1}^{(k)},1/ A_{2i}^{(k)}\big ]\). They also obtained the values of \(A_i^{(k)}\) for \(i=2,3,4\). In this note, following Nathanson and Bryant’s work, we further determine the value of \(A_5^{(k)}\).

令\（{\ mathbb {N}} \）为正整数，而\（k \ ge 3 \）为整数。定义集合\（G \ subseteq（0,1] \）的实数作为一个 ķ -good集合如果ģ包含长度的无几何级数ķ一些比\（R \在{\ mathbb {N}} \ setminus \ {1 \} \） 。实数\（X \在（0,1] \ setminusģ\）是ķ -bad相对于 ģ如果存在一个整数在\（R \ {\ mathbb {N} } \ setminus \ {1 \} \）使得\（G \ cup \ {x \} \）包含k个项的几何级数\（（x，xr，xr ^ 2，\ ldots，xr ^ {k-1}）\）。定义\（Bad（G）= \ {x \ in（0,1] \ setminus G：x \ quad \ text {是} \ quad k \ text {-bad} \ text {与} \ text {尊重} \ text {to} \ quad G \} \）。2015年，Nathanson和O'Bryant（Acta Arith 170：327–342，2015）证明存在一个唯一的严格递增的序列\（\ big \ {A_1 ^ {（ k）} <A_2 ^ {（（k）} <\ cdots \ big \} \）带有\（A_1 ^ {（k）} = 1 \）的正整数，这样\（G ^ {（k）} = \ bigcup _ {i = 1} ^ {\ infty} \ big（1 / A_ {2i} ^ {（k）}，1 / A_ {2i-1} ^ {（k）} \ big] \）是一个k -好集和\（Bad（G ^ {（k）}）= \ bigcup _ {i = 1} ^ {\ infty} \ big（1 / A_ {2i + 1} ^ {（k）}，1 / A_ {2i} ^ {（k）} \ big] \），他们还获得了\（A_i ^ {（k）} \）的值\（i = 2,3,4 \）。在此笔记中，根据Nathanson和Bryant的工作，我们进一步确定\（A_5 ^ {（k）} \）的值。