Most "compact" sequence such that there is at least one prime between a(n) and a(n+1)

[franktaw at netscape.net]

I believe it has been proved that there is always a prime between n^3 
and (n+1)^3.  So the cubes are a considerably more compact example.

Franklin T. Adams-Watters

-----Original Message-----
From: Andrew Plewe <aplewe at sbcglobal.net>

I'd consider that valid, but it's by no means the most "compact:

1, 2, 3, 7, 43, 1892,...

While looking up Andrica's conjecture, I found this note:

5) p   / p  <= 5/3, and the maximum occurs at n = 2.
    n+1   n
  (This last conjecture has been proved to be true by Jozsef Sandor,
   Babes-Bolyai University of Cluj, "On A Conjecture of Smarandache on
    Prime Numbers", <Smarandache Notions Journal>, Vol. 10, 2000, to 
appear.)

Reference: http://www.gallup.unm.edu/~smarandache/conjprim.txt


I checked the reference and the paper was published. I assume, 
therefore,
that the proof is accepted. Let p_n = 2 and maximize the sequence by 
setting
p_n+1/p_n = 5/3. Proceed as follows:

a(1) = 2
a(2) = x/2 = 5/3, by cross-multiplying: 3x = 10, then ceiling(10/3) = 4
a(3) = x/4 = 5/3, 3x = 20, ceiling(20/3) = 7
a(4) = x/7 = 5/3, 3x = 35, ceiling(35/3) = 12
a(5) = x/12 = 5/3, 3x = 60 = 20
a(6) = x/20 = 5/3, 3x = 100, ceiling(100/3) = 34
a(7) = x/34 = 5/3, 3x = 170, ceiling(170/3) = 57
a(8) = x/57 = 5/3, 3x = 285 = 95

and so on. This is ultimately, I believe, more "compact" than A052548:

3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026...

If I've constructed the sequence correctly, than their is either a prime
between a(n) and a(n+1), or a(n+1) is prime.

-----Original Message-----
From: Stefan Steinerberger [mailto:stefan.steinerberger at gmail.com]
Sent: Tuesday, December 11, 2007 12:45 PM
To: Andrew Plewe
Cc: Robert Israel; Jonathan Post; seqfan at ext.jussieu.fr; jonathan post;
maximilian.hasler at gmail.com
Subject: Re: Most "compact" sequence such that there is at least one 
prime
between a(n) and a(n+1)

> Thus a clarification; generating the sequence shouldn't
> require that we know which numbers are prime.

That's a bit vague, is it not? Let's define a(1) = 2 and
a(n+1) as the smallest number that can't be written
as the product of numbers in {1,2,3,...,a(n)} plus 1.
That way we should always get a prime in between
without knowing what primes are.

An almost number-theoretic free sequence could be defined
as such: Andrica's conjecture implies p_(n+1) - p_(n) < 2*sqrt(p_n) +1.
Assuming its truth and defining a(1) = 2 and
a(n+1)  = ceiling(a(n) + 2*sqrt(a(n)) + 1) should give such
a sequence. (Are there any nice proven bounds like that?)

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Tom,   let me just make a small comment - i get a lot of email,
roughly 1 every 2 minutes.

Some I process on the fly, some - most - get put into
the appropriate queue (comments, new sequences, etc.) to
be processed in batches, but possibly a few weeks after 
being received

your sequences, i will of course be very grateful for the
corrections, but i might also wish that you got things right
the first time!  Wish that you carefully composed the sequence
or comment in a file, double-checked it, and then sent it 
to me.

In short, if you send 2 messages about a sequence, I might
not read them in the right order!

I'll copy this to the seqfan list

Neil


PS Another complication is that emails have been getting lost
- if you send me a message, or submit a sequence or comment
using the web page, always keep a comment.