[seqfan] Is there a phenomenon?

Vladimir Shevelev shevelev at bgu.ac.il
Sat Sep 17 22:59:17 CEST 2011

For a real r>1, let us call a prime p an r-gap prime, if there is no prime between r*p and r*q, where q is the next prime after p. A few the first terms of 3-gap primes are 
71, 107, 137, 281, 347, 379, 443, 461, 557, 617, 641, 727, 809, 827, 853,..., 
while the first terms of 2.5-gap primes are
127, 197, 281, 311, 347, 431, 613, 659, 673, 739, 877, 991, 1049, 1229, 1277.
Looking at these tables I did a conjecture that always a_3(n)<=a_2.5(n), where the equality holds only for n=5. But, in cases when a_3(n)<5/6*a_2.5(n) this means that the longer prime gaps of a certain form appear earlier than the shorter ones. Thus, if the inequality a_3(n)<5/6*a_2.5(n) occur more often than the opposite one, then we have a phenomenon.
Can anyone verify such inequality for larger n and, maybe, disprove this phenomenon?

Best regards,

 Shevelev Vladimir‎

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