Školjka

%V0 Let $\alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $n$. Prove that: (a) the inequality $\alpha(n) \left(n^2 \right) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds; (b) the above inequality is an equality for infinitely many positive integers, and (c) there exists a sequence $\left(n_i \right)^{\infty}_1$ such that $\frac{\alpha \left( n^2_i \right)}{\alpha \left(n_i \right)}$ goes to zero as $i$ goes to $\infty$. Alternative problem: Prove that there exists a sequence a sequence $\left(n_i \right)^{\infty}_1$ such that $\frac{\alpha \left( n^2_i \right)}{\alpha \left(n_i \right)}$ (d) $\infty$; (e) an arbitrary real number $\gamma \in (0,1)$; (f) an arbitrary real number $\gamma \geq 0$; as $i$ goes to $\infty$.