[seqfan] Re: On some constellations of primes

Sun Oct 14 19:03:04 CEST 2012

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/