[seqfan] Re: On some constellations of primes

[Vladimir Shevelev]

Thank you, Hans, for this right and important remark. Such sets of consecutive primes are connected with the isolated primes (A166251). It is based on Propositions 13 and 16 of my paper in link. These propositions
forbid to the interior primes of such a sequence to be non-isolated, but allow to the first prime to be only "isolated from the right", while to the last prime to be only "isolated from the left"
(or, by my classification, the first prime can be "left prime", while  the last prime can be "right prime").  Therefore, in the constructing the suggested
sequence we need to verify one prime before a run of consecutive isolated primes and one prime after it. I hope that you (or you and Zak) can submit this sequence. 
 
Best,
Vladimir


----- Original Message -----
From: Hans Havermann <gladhobo at teksavvy.com>
Date: Tuesday, October 9, 2012 1:46
Subject: [seqfan] Re: On some constellations of primes
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> Vladimir Shevelev:
> 
> > The following sequence of 11 consecutive primes
> > 55469,55487,55501,55511,55529,55541,55547,55579,55589,55603,55609
> > possesses an interesting property: between every adjacent half-
> 
> > primes there exists at least one prime. In particular, between 
> the  
> > first two half-primes there are 3 primes: 27737,27739,27743.
> 
> 
> The prime previous to 55469 is 55457. Between 55457/2 and 
> 55469/2 is  
> the prime 27733.
> The prime after 55609 is 55619. Between 55609/2 and 55619/2 is 
> the  
> prime 27809.
> 
> I don't understand why the two either-end consecutive primes are 
> being  
> excluded here. This appears to be so as well for Vladimir's 
> follow-up  
> "a(2)=5, a(3)=79, a(4)=541, a(5)=6599, a(6)=10771".
> 
> 
> Zak Seidov:
> 
> > Smallest set of 13 (VladSh's) consecutive primes:
> > s=prime(1785277..1785289)={28751809, 28751851, 28751857, 
> 28751873,  
> > 28751893, 28751903, 28751929, 28751941, 28751969, 
> 28751977,  
> > 28752007, 28752019, 28752037},
> > 12 corresponding smallest primes q(k) between (1/2)s(k) and 
> (1/2)s(k 
> > +1):
> > q(k=1..12)={14375923, 14375927, 14375929, 14375939, 
> 14375947,  
> > 14375957,
> >  14375969, 14375981, 14375987, 14376001, 14376007, 14376013};
> 
> The prime previous to 28751809 is 28751773. Between 28751773/2 
> and  
> 28751809/2 is the prime 14375899.
> I'm going to guess that Zak's program searched for 13 intervals 
> (i.e.,  
> 14 consecutive primes).
> 
> 
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> 

 Shevelev Vladimir‎