Let \(s(n)\) denote the sum of digits in the binary expansion of the integer \(n\). K. G. Hare et al. [Int. J. Number Theory 7, No. 7, 1737–1752 (2011; Zbl 1270.11008)] studied the number of odd integers such that \(s(n) = s(n^2 ) = k\), for a given integer \(k\ge 1\). They settled all cases with the exceptions of \(k\in\{9, 10, 11, 14, 15\}\). The main motivation to consider this kind of question comes from a work of M. Madritsch and T. Stoll [Acta Math. Hung. 143, No. 1, 192–200 (2014; Zbl 1333.11010)] who showed that \((s(n^2 )/s(n))_{n\ge 1}\) is dense in \(\mathbb{R}^+\).
In this paper, the authors 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 \(k\in\{14, 15\}\) (where they conjecture that the number of odd integers \(n\) with \(s(n) = s(n^2 ) = k\) is also finite).
A related problem is to study the set \(E_4\) of solutions of \(s(n^2 ) = 4\) for odd integers. M. A. Bennett et al. [Math. Proc. Camb. Philos. Soc. 153, No. 3, 525–540 (2012; Zbl 1291.11016)] proved that there are only finitely many solutions and conjectured that \(E_4=\{13, 15, 47, 111\}\). In this paper, the authors provide an algorithm to find all solutions with a fixed sum of digits value \(\lambda\). Supporting Bennet et al. conjecture, they show that \(\cup_{\lambda\le 17} E_{4,\lambda}=\{13, 15, 47, 111\}\) where \(E_{k,\lambda}=\{n\mid s(n^2 ) =k,\ s(n)=\lambda\}\). They obtain as well related results for \(s(n^2 ) = 5\).