2*p_(i-n) > p_i

for i > k where k = primepi(p_k) = primepi(R_n). That is, p_k is the n'th Ramanujan Prime, R_n, and the k'th prime.

Proof:

One can rewrite S. Ramanujan's paragraph 2. of "A proof of Bertrand's postulate" to the above. (link: http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper24/page1.htm )

Example:

From T. D. Noe's table's (links at:

http://www.research.att.com/~njas/sequences/A104272 ,

http://www.research.att.com/~njas/sequences/A000720 )

p_k = 19403, k = 2197, with n=1000, therefore i >= to 2198 and i-n >= 1198. The 2198th prime is 19417, and the 1198th prime is 9719. 2*9719 = 19438 > 19417.

enjoy,

John W. Nicholson