MathDB
S 15

Source:

May 25, 2007
inequalitieslimitinductionMiscellaneous Problems

Problem Statement

Let α(n)\alpha(n) be the number of digits equal to one in the dyadic representation of a positive integer nn. Prove that [*] the inequality α(n2)≤12α(n)(1+α(n))\alpha(n^2 ) \le \frac{1}{2} \alpha(n) (1+\alpha(n)) holds, [*] equality is attained for infinitely n∈Nn\in\mathbb{N}, [*] there exists a sequence {ni}\{n_i\} such that lim⁡i→∞α(ni2)α(ni)=0\lim_{i \to \infty} \frac{ \alpha({n_{i}}^2 )}{ \alpha(n_{i}) } = 0.
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