Answer
$$(\dfrac{13}{6}, \dfrac{15}{6}, \dfrac{20}{6}, \dfrac{25}{6}, \dfrac{33}{6}, \dfrac{44}{6})$$
Work Step by Step
$$\vec{z}=\dfrac{\overrightarrow{N}}{3}+ \dfrac{\overrightarrow{N}}{2}\\=\dfrac{(5,6,7,8,9,10)}{3}+\dfrac{ (1,1,2,3,5,8)}{2} \\=(\dfrac{10+3}{6}, \dfrac{12+3}{6},\dfrac{14+6}{6}, \dfrac{16+9}{6}, \dfrac{18+15}{6}, \dfrac{20+24}{6})\\=(\dfrac{13}{6}, \dfrac{15}{6}, \dfrac{20}{6}, \dfrac{25}{6}, \dfrac{33}{6}, \dfrac{44}{6})$$