The sequence $\{f_n\}$ starts with an index of $1$ and is defined so that $f_n$ is the largest integer $k$ such that $k^{2}n.$ Which sequence fits the definition of $\{f_n\}?$

$1,1,2,2,$dots

$1,2,3,4,5,$dots

$1,4,9,16,25,$dots

$2,4,8,16,32,$dots