For every fixed $n \in \mathbb{N}$, Rofl Brandl and Shi Wujie gave in *Finite groups whose elements are consecutive integers* (Journal of Algebra, **143**, 388-400 (1991).) a complete classification of finite groups whose spectrum is $\{1,2,\ldots,n\}$. A particularly appealing spin-off of their study is the following one:

Let $i$ be a positive integer greater than $8$. There is no finite group $G$ whose spectrum is $\{1,2,\ldots,i\}$.