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Aaron,your method of the semi-greedy strategy (greedy from a number and over ) is interesting and i want to thank you very much for your efforts and your interest ,but the question whether the non-sieved numbers from these greedy methods grow faster than the primes is ,as it seems, too sharp to be answered analytically(?). 2^40 >> 110,000 so...
@S. Carnahan:i am not convinced , how many numbers until this are out of the cover? compare it with the 2^k (where k is this number). i think that the question remains full open
Westzynthious article admits that you can have "big" intervals without using many primes. This could be a problem for claims for large numbers but i am not sure , i am working on it.... mathoverflow.net/questions/37679/…
