[seqfan] Re: A020497 name needs improvement

Vladimir Shevelev shevelev at bgu.ac.il
Tue Jun 4 13:15:27 CEST 2013

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."

----- 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.
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> Seqfan Mailing list - http://list.seqfan.eu/

 Shevelev Vladimir‎

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