[seqfan] Re: On some constellations of primes

[Vladimir Shevelev]

One can submit the following sequence in OEIS:
"a(n) is the least prime of the set of the smallest n consecutive primes a(n)=q_1(n), q_2(n),..., such that between (1/2)*q_i  and (1/2)q_(i+1), i=1,...,n-1, there exists a prime."
We have a(2)=5, a(3)=79, a(4)=541, a(5)=6599, a(6)=10771,...
The sequence, generally speaking, is not monotonic. For example, according
to Zak's excellent calculations, a(12)=28751851>a(13)=28751809. 
 
Best regards,
Vladimir
 


----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Friday, October 5, 2012 6:59
Subject: [seqfan] On some constellations of primes
To: seqfan at list.seqfan.eu

> Dear SeqFans,
>  
> 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. 
> This sequence is connected with two my sequences:  A166251 
> and A182426 (see a theorem in comment). I conjectured that there 
> exist arbitrary long sequences of consecutive primes with such 
> property. Reinhard Zumkeller extended A182426 up to n=10000 and 
> after that (private communication) up to n=50000, but I did not 
> find a term>11. Thus till now I do not know a longer sequence 
> than the above mentioned one. Can anyone find a longer sequence 
> or indicate
> a new bound N, such that up to N such a sequence not exists.
>  
> Regards,
> Vladimir
> 
>  Shevelev Vladimir‎
> 
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 Shevelev Vladimir‎