Lukaluce wrote: We say that a positive integer $n$ is memorable if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares? Yes, there are . Choose for example $(2^n-5)^2$ where $n\ge 5$ : this perfect square has $n+1$ digits $1$ and $n-1$ digits $0$ in its binary representation.

$n^2=2^k \cdot a_k + ... + 2^1 \cdot a_1 + 2^0 a_0$ Next number$$\boxed {(2^{k+2} + 1)n}$$