  Let $s(n)$ denote the sum of digits in the binary expansion of the integer
$n$. Hare, Laishram and Stoll (2011) studied the number of odd integers such
that $s(n)=s(n^2)=k$, for a given integer $k\geq 1$. The remaining cases that
could not be treated by theses authors were $k\in\{9,10,11,14,15\}$. In this
paper we show that there is only a finite number of solutions for
$k\in\{9,10,11\}$ and comment on the difficulties to settle the two remaining
cases $k\in\{14,15\}$. A related problem is to study the solutions of
$s(n^2)=4$ for odd integers. Bennett, Bugeaud and Mignotte (2012) proved that
there are only finitely many solutions and conjectured that $n=13,15,47,111$
are the only solutions. In this paper, we give an algorithm to find all
solutions with fixed sum of digits value, supporting this conjecture, as well
as show related results for $s(n^2)=5$.
