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 notable consequence of their main theorem is the following one:

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