[seqfan] Re: On some constellations of primes

Vladimir Shevelev shevelev at bgu.ac.il
Mon Oct 15 00:20:55 CEST 2012 

(A continuation after the sketch). Let now m=m(n). Roughly we have 
2^{n/m(n)}<=n*ln(n), or m(n)>=n/log_2(n*ln(n))=n*ln2/ln(n*ln(n)).
Since a(n) is close to maximal integer<=n*ln(n), then we accept roughly that
the number of considered primes not exceeding n is n*ln2/ln(n*ln(n)). It is interesting that this formula predicts very good the exact number of such primes (cf. list of them in A217691 extended by Alois).  So we have (the first number to the right is the predicted number of primes and in brackets the exact number).
 
n=100: 11.3 (10)
n=200: 19.9 (19)
n=300: 27.9 (26)
n=400: 35.6 (34)
n=500: 43.0 (41)
n=600: 50.4 (48)
 
Can anyone verify this approximation for larger n?
 
Regards,
Vladimir



----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Friday, October 12, 2012 22:03
Subject: [seqfan] Re: On some constellations of primes
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> 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/
> 

 Shevelev Vladimir‎

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