[seqfan] Re: Is there a phenomenon?

[zak seidov]

Vlad,
"conjecture that always a_3(n)<=a_2.5(n)" is wrong.
Zak 

----- Original Message -----
> From: Vladimir Shevelev <shevelev at bgu.ac.il>
> To: seqfan at list.seqfan.eu
> Cc: 
> Sent: Saturday, September 17, 2011 11:59 PM
> Subject: [seqfan] Is there a phenomenon?
> 
> 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,
> Vladimir
> 
> 
> Shevelev Vladimir‎
> 
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