Abstract :
In 1954 Lorentz and Erdös showed that there are very thin sets of positive integers complementary to the set of primes. In particular, there is anAsubset of or equal toimage with[formula]and[formula]Erdös conjectured that the bound (i) could be sharpened too(ln2 x) or evenO(ln x). In the present paper it is proved that there are setsAwith[formula]such that (ii) is fulfilled for almost alln.
