[seqfan] Re: An interesting sequence connected with primes

[Charles Greathouse]

I feel this is related to the behavior of A005250 but I don't have the time
at the moment to pin down the relationship.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Thu, Sep 26, 2013 at 11:34 AM, Vladimir Shevelev <shevelev at bgu.ac.il>wrote:

>
> 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
> > >
> > > _______________________________________________
> > >
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> > >
> >
> > _______________________________________________
> >
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> >
>
>  Shevelev Vladimir
>
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