Abstract: Let $G$ be a finite group and $d$ a positive integer. Let $s_{d\mathbb{N}} (G)$ denote the smallest positive integer $l$ such that every sequence over $G$ of length at least $l$ contains a nonempty product-one subsequence $T$ with $|T|\equiv0 \pmod{d}$. This paper studies $s_{d\mathbb{N}} (D_{2n})$ for the dihedral group $D_{2n}$ and shows that when $n=2^r$ with $r\geq3$, the equality $s_{d\mathbb{N}} (D_{2n})=\operatorname{lcm}(n, d)+\gcd(n, d)$ holds.

Key words: Dihedral group, Product-one sequence, Congruence condition, Davenport constant