EOJ Monthly 2018.8

In the beginning, there were no primes, or numbers. And God said, “let there be numbers,” and there were numbers. God thought that it might be a beautiful thing to have some of these numbers added up, and he called some of the sums “primes”.

Some scholars argued that there should be no more than $8$ numbers God first created, as the largest prime written in these ancient books was $233$. But others found this not convincing, as it was debatable whether $233$ was a prime.

If the legend somehow confuses you, here is a more formal version:

Find out the smallest positive integer $k$ such that there is a set (non-repeated) of positive integers $X$ where $|X|=k$, and any prime number in $1,2,\dots ,n$ can be expressed as the sum of a subset of $X$.