A primitive sequence $1<a_1<\ldots<a_k\leq n$ is a sequence of integers no one of which divides any other, investigated by Erdos, Behrend and others, over the last 80 years. In fact, $\max k=\lfloor \frac{n+1}{2}\rfloor,$ which can be shown by taking exactly one member from each chain in the divisibility poset, namely $$C_x=\{x,2x,2^2x,\ldots\} \subset \{1,\ldots,n\}$$as $x$ ranges over the odd integers in $\{1,\ldots,n\}$. The easy examples of such sets, e.g., the numbers in $\{1,\ldots,n\}\cap [\lceil \frac{n}{2} \rceil,n]$ have the following property $$\sum_{i:a_i\leq n} \frac{1}{a_i} \leq c$$ for some absolute constant $c$ as $n$ increases.

However, Erdos states that S. Pillai observed that there is a constant $c'$ such that for every $n$ there is a finite primitive sequence all whose terms are upper bounded by $n$ such that $$\sum_{i:a_i\leq n} \frac{1}{a_i} > c' \frac{\log n}{\sqrt{\log \log n}}.$$

Are there any results on what such sequences may look like?

**Edit:** Is the value of the constant $c'$ or bounds on it known?