[seqfan] Re: On some constellations of primes

Sat Oct 13 02:22:45 CEST 2012

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