I have weird and vague question. We know the reciprocal of numbers
$$\sum_{k\leq n}\frac{1}{k}\sim \log n$$
and reciprocal of primes
$$\sum_{p\leq n}\frac{1}{p}\sim \log\log n$$
Now consider reciprocal of some sort of primes
$$\sum_{p*\leq n}\frac{1}{p*}\sim \log\log\log n$$
where $p*$ is the element of a subset of the prime number set. What would $p*$ be?