For every fixed $n \in \mathbb{N}$, Rolf Brandl and Shi Wujie gave in Finite groups whose element orders 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 $\{1,2,\ldots,i\}$.
