On simultaneous binary expansions of $n$ and $n^2

Journal of Number Theory, 111 (2005), 248 -- 256 A new family of sequences is proposed. An example of sequence of this family is more accurately studied. This sequence is composed by the integers $n$ for which the sum of binary digits is equal to the sum of binary digits of $n^2$. Some structure and asymptotic properties are proved and a conjecture about its counting function is discussed.