[seqfan] Re: An interesting sequence connected with primes

[Vladimir Shevelev]

Thank you, Charles, it is a good base for a  deeper explanation. Here  there are many interesting cases.
Let, for example, prime(n) and prime(n+1) be twin primes. Then, by the condition, we easily find 
 a(n+1) - a(n) + 1 = 1/2 *(nextprime(2*a(n+1)+1) - nextprime(2*a(n)+1))
and, by observations, "as a rule" a(n+1) - a(n) = -1. Then  nextprime for 2*a(n)+1 and 2*a(n+1)+1 is common.
Let us call such twin primes "regular". But there are also cases when a(n+1)>a(n). These cases correspond to "irregular" 
twin primes. The first ones are  (3,5), (5,7), (149,151).
 
Best regards,
Vladimir


----- Original Message -----
From: Charles Greathouse <charles.greathouse at case.edu>
Date: Wednesday, September 25, 2013 2:20
Subject: [seqfan] Re: An interesting sequence connected with primes
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> Those are instances using the same prime. The next prime after 
> 2*58+1 is
> 127, which is also the next prime after 2*57+1. These are within 
> 1 because
> 11 is within 2 of 13.
> 
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
> 
> 
> On Wed, Sep 25, 2013 at 9:16 AM, Vladimir Shevelev 
> <shevelev at bgu.ac.il>wrote:
> > Dear SeqFans,
> >
> > I would like to discuss an astonishing behavior of the 
> following sequence
> > connected with primes ( I submitted it as A229512):
> > a(n) is the minimal k, such that nextprime(2*k+1)-
> 2*k=prime(n). It begins
> >
> >
> > 
> 0,1,3,11,58,57,262,261,564,666,665,4775,7843,7842,9807,9804,15705,15704,15701,15699,15698,>
> > 
> 77964,77962,180330,180326,185136,185135,185133,185132,185130,678603,678601,1005372,> 1005371,1005366,2326178,8525865,8525862,...
> >
> > It is interesting the distribution of terms over gropes of close
> > magnitudes. For example, 58,57; 9807,9804;, 
> 15704,15701,15699,15698; etc.
> > Could anyone to give an explanation?
> >
> > Best regards,
> > Vladimir
> >
> >  Shevelev Vladimir
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/
> 

 Shevelev Vladimir‎