[seqfan] Re: On some constellations of primes

Sun Oct 14 20:44:14 CEST 2012

There is a sequence with 16 terms starting in the prime 784284211 and
ending in the prime 784284637

Carlos Rivera

On Sun, Oct 14, 2012 at 12:03 PM, Charles Greathouse <
charles.greathouse at case.edu> wrote:

> Carlos Rivera sends (via J. M. Bergot) an example with 15 primes:
> 278689963
> 278689991
> 278690017
> 278690039
> 278690053
> 278690131
> 278690183
> 278690249
> 278690297
> 278690329
> 278690359
> 278690369
> 278690393
> 278690443
> 278690519
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
> On Fri, Oct 12, 2012 at 8:22 PM, Vladimir Shevelev <shevelev at bgu.ac.il>
> wrote:
> > The following sketch shows that the conjecture is true. Suppose that the
> sequence contains a finite number (m) primes: q_1,...,q_m. Then every term
> > has the form q_i *2^k, i<=m, and every new term appears by
> multiplication of one of the previous term by 2. We have "multigeometric"
> progression based on several (m) bases {q_i}. It is not difficult to prove
> that its maximum grows not slowly than
> > 2^{n/m} for n>=n_0, and, if to try to forbid such a growth for the
> maximum, then we should multiplicate by 2 before-maximum,
> before-before-maximum,...,minimum, such that iniformly all terms of the
> considered progression grow not slowly, i.e., a(n) grows not slowly than
> 2^{n/m}. It is contradiction, since 2^{n/m}>>n*ln(n). Thus the sequence
> contains infinitely many primes and limsup a(n)/(n*ln(n))=1. The question
> on liminf I remain open. I think that if to replace 2 by e, then, possibly,
> also liminf a(n)/(n*ln(n))=1.
> >
> > Regards,
> > Vladimir
> >
> >
> >
> > ----- Original Message -----
> > From: Vladimir Shevelev <shevelev at bgu.ac.il>
> > Date: Friday, October 12, 2012 3:36
> > Subject: [seqfan] Re: On some constellations of primes
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> >
> >> Dear SeqFans,
> >>
> >> I was interested also in the maximal sequence {a(n)} with the
> >> condition a(n)<=prime(n) for which NO terms of the sequence
> >> between a(n)/2
> >> and a(n+1)/2. I obtained sequence A217689, all terms of which
> >> have the form 2^k*p, where k>=0, p is prime>=2. The sequence
> >> hypothetically  contains infinitely many primes:
> >> 2,3,19,23,31,43,59,61,73,83,103, etc. If anyone can look at it
> >> by a "fresh glance", maybe, this conjecture is not so deep?
> >> Note that a close idea could be used to many other sequences.
> >> For example, if the primes to replace by the squares, then, for
> >> n>=0, we obtain the sequence  A217833:
> >> 0,1,2.4,8,16,32,49,64,81,98,121,128,162,196,242, etc., the terms
> >> of which have the form s*2^k, where s>=0 is a square.
> >>
> >> Regards,
> >> Vladimir
> >>
> >>
> >> ----- Original Message -----
> >> From: Vladimir Shevelev <shevelev at bgu.ac.il>
> >> Date: Wednesday, October 10, 2012 3:37
> >> Subject: [seqfan] Re: On some constellations of primes
> >> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> >>
> >> > Now this sequence is A217671. After the replacing "run" by
> >> "set"
> >> > it is, evidently, monotonic. I invite colleagues to verify and
> >> > continue it.
> >> >
> >> > Regards,
> >> > Vladimir
> >> >
> >> >
> >> >
> >> > ----- Original Message -----
> >> > From: Vladimir Shevelev <shevelev at bgu.ac.il>
> >> > Date: Tuesday, October 9, 2012 6:57
> >> > Subject: [seqfan] Re: On some constellations of primes
> >> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> >> >
> >> > > 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).
> >> > > >
> >> > > >
> >> > > > _______________________________________________
> >> > > >
> >> > > > Seqfan Mailing list - http://list.seqfan.eu/
> >> > > >
> >> > >
> >> > >  Shevelev Vladimir
> >> > >
> >> > > _______________________________________________
> >> > >
> >> > > Seqfan Mailing list - http://list.seqfan.eu/
> >> > >
> >> >
> >> >  Shevelev Vladimir
> >> >
> >> > _______________________________________________
> >> >
> >> > Seqfan Mailing list - http://list.seqfan.eu/
> >> >
> >>
> >>  Shevelev Vladimir
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
> >  Shevelev Vladimir
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
>

-- 
C.Rivera
www.primepuzzles.net
cbrfgm at gmail.com