Let's denote the sequence as $A$.

The numerical sequence is defined by the conditions: $a_{1}=1, a_{n+1}=a_{n}+\left[\sqrt{a_{n}}\right]$.

How many perfect squares are there among the first terms of this sequence, not exceeding $1000000$?
According to the solution of problem $\underline{98152}$, all perfect squares in this sequence have the form $4^{m}$. There are exactly 10 numbers of this form within the specified limits

$\left(4^{10}=1024^{2}>10^{6}\right)$.