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 {$1,2,\ldots,i$}.