The fiftieth non-square number

Text

Consider the sequence of positive integers that are not perfect squares: $2,3,5,6,7,8,10,\dots$ (skipping $1,4,9,16,\dots$). What is the $50$-th term?

Solution

Up to a number $N$ the perfect squares are $\lfloor\sqrt N\rfloor$. We look for $N$ such that the non-squares up to $N$ are exactly $50$, i.e. $N-\lfloor\sqrt N\rfloor=50$. Try $N=57$: the squares $\le 57$ are $1,4,9,16,25,36,49$, that is $7$; then the non-squares $\le 57$ are $57-7=50$, and $57$ is not a square. Hence $a_{50} = \boxed{57}.$ (In general the $k$-th non-square is $k+\text{round}(\sqrt k)$: $50+\text{round}(\sqrt{50})=50+7=57$.)