Putnam 1991 B1

For each integer $n \geq 0$, let $S(n) = n - m^2$, where $m$ is the greatest integer with $m^2 \leq n$. Define a sequence $(a_k)_{k=0}^\infty$ by $a_0 = A$ and $a_{k+1} = a_k + S(a_k)$ for $k \geq 0$. For what positive integers $A$ is this sequence eventually constant?