The Jacobsthal function applied to the product of the first n primes (see OEIS sequence A048670:

http://oeis.org/search?q=A048670&sor...lish&go=Search) provides an answer for ranges up to 742. For example, to get a range of 210 covered it takes 23 primes (i.e. up to 83).

The assumption that a covering set consists only of the primes in consecutive order has also proven to be false.