# Maximum length sub-sequence of probability reciprocals

Find a non-ascending sequence of non-zero probabilities $$p_1,\dots,p_N$$ s.t.

1. the sum of probabilities is $$1$$ (i.e., a distribution), and
2. maximizes the length of the sub-sequence $$1/p_1,\dots,1/p_R$$ where the sum is no greater than $$N$$.

Formally,

maximize $$R$$

s.t. $$\sum_{i=1}^{N} p_i$$ = 1 and $$\sum_{i=1}^{R} 1/p_i \leq N$$.

Observations:

1. If every $$p_i=1/N$$ then $$R=1$$.

2. If $$p_i= (1-\varepsilon) / \sqrt{N}$$ for $$1 \leq i \leq \sqrt{N}$$, and $$p_i=\varepsilon/(N-\sqrt{N})$$ for $$\sqrt{N}+1 \leq i \leq N$$, then $$R=(1-\varepsilon)\sqrt{N}$$. Is this a maximum for $$R$$?

3. Based on this answer, can we say that when $$R$$ is maximized, all $$p_i$$'s in the sub-sequence are equal?

I found an answer that very closely addresses the problem. Based on that, it is easy to see that observation 2 actually gives the maximum for $$R$$.