[seqfan] Re: A020497 name needs improvement

Vladimir Shevelev shevelev at bgu.ac.il
Fri Jun 7 11:18:41 CEST 2013


If to understand A020497 in David's sense: "A020497(n) gives smallest k such that k  consecutive integers admits a permissible prime pattern of size n", then it is easy to see that for n>=3 a(n)<=(A080359(n)+1)/2. This estimate is better than a(n)<=R(n)/2. On the other hand, prime(n+2)-4 is a very good approximation of a(n). For example, a(n)=prime(n+2)-4
for n=1,2,3,4,5,12,13,15,17,21,35,36,37,40,41, etc. and I believe that this sequence is infinite.

Regards,
Vladimir


----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Monday, June 3, 2013 23:24
Subject: [seqfan] Re: A020497 name needs improvement
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> I quite agree with David. For example,  if y=a(2)=3, then, 
> for integer x, the set {x+1, x+2, x+3} should contain twin 
> primes {x+1, x+3} for infinitely many x which is a best known 
> unsolved problem. The name suggested by David seems to be suitable.
> On the other hand, even for the existing name, a(n) exists. For 
> example, a(n)<=R(n)/2, where R(n) is the n-th Ramanujan 
> primes (A104272). So, another suitable  name is 
> "Hypothetically minimal values of y such that n primes occur 
> infinitely often among (x+1, ..., x+y), i.e. pi(x+y)-pi(x) >= n 
> for infinitely many x."
>  
> Regards,
> Vladimir
> 
> 
> ----- Original Message -----
> From: David Wilson <davidwwilson at comcast.net>
> Date: Monday, June 3, 2013 17:41
> Subject: [seqfan] A020497 name needs improvement
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> 
> > I think the description of A020497 is incorrect as it is 
> conjectural.> 
> >  
> > 
> > I believe that A020497(n) gives smallest k such that k 
> > consecutive integers
> > admits a permissible prime pattern of size n.
> > 
> >  
> > 
> > The first Hardly-Littlewood conjecture (the k-tuples 
> conjecture) 
> > impliesthat an admissible prime pattern is satisfied by an 
> > infinite number of prime
> > constellations (which implies there are an infinite number of 
> > values x with
> > pi(x+a(n)) - p(x) = n). However, this is still conjectural, 
> and 
> > has not been
> > proved even for the admissible pattern (0,2), which is to say, 
> > we have not
> > yet proved the twin prime conjecture. For sufficiently large k 
> (where> sufficiently large is not very large), we can neither 
> > demonstrate nor prove
> > that a single prime constellation satisfies a maximal 
> admissible prime
> > pattern of size n.
> > 
> >  
> > 
> >  
> > 
> > 
> > _______________________________________________
> > 
> > Seqfan Mailing list - http://list.seqfan.eu/
> > 
> 
>  Shevelev Vladimir‎
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/
> 

 Shevelev Vladimir‎



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