TITLE: Finding minimum value of $1/(p_1n_1) + 1/(p_2n_2) + 1/(p_3n_3) + \dots + 1/(p_kn_k)$
QUESTION [5 upvotes]: While solving a statistic problem, I come across an equation:
The value of $p_1$ is given, and $p_{i+1} = p_i / (1-(1-p_i)^{n_i})$.
$$E = p_1  (\frac{1}{p_1  n_1} + \frac{1}{p_2 n_2} + \dots + \frac{1}{p_{k-1}  n_{k-1}} + \frac{1}{p_k  n_k})
= p_1  \sum_i\frac{1}{p_i n_i},$$
where $i = 1, \dots, k$, $0<p_i<1$, $n_i \in N$, $n_k = 1$.
The task is to (1) find the number $k$, and (2) find each $n_i$, such that $E$ has minimum value.
After some brute-force searches with a computer program I wrote, it seems that the $n_i$ and $p_i$ that give the minimum value to $E$ have the property: $n_i/n_{i+1}≈2$ and $p_{i+1}/p_i≈2$, except the last term. This is all I've got so far.
Any thought on solving this problem other than brute-force searches?
Thank you.
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Below are some results of the brute-force searches. For $p_1$>0.01, this program works well to find the best combination of $n_i$ and $k$. However, for $p_1$<0.01, it will be too much work to exhaust the search.

REPLY [3 votes]: I've developed a calculator to find the combination of $n_i$ that gives $E$ the minimum value. The algorithm will give some approximately "good enough" solution, but not always the best solution, to this question.
https://shorturl.at/fixO5