question about A033188 (Conjectural) minimum difference of an increasing arithmetic progression of n primes.

[David Wilson]

If the smallest element of an arithmetic progression of n primes (that is, 
A033189(n)) is > n#, then modular arguments show that the difference (that is, 
A033188(n)) must be >= n#.  A variation of the k-tuples conjecture then asserts 
that A033188(n) = n#.  The empirical data on A033189 seem to indicate that 
A033189(n) >= n# for n >= 8.

The proved values for A033189(n) will be those for which A033189(n) are known.

Can someone please verify A033189(13).  I would have expected A033189 to be 
increasing, but it ain't necessarily so.

----- Original Message ----- 
From: "N. J. A. Sloane" <njas at research.att.com>
To: <seqfan at ext.jussieu.fr>
Cc: <njas at research.att.com>
Sent: Thursday, January 26, 2006 3:07 PM
Subject: question about A033188 (Conjectural) minimum difference of an 
increasing arithmetic progression of n primes.


> I'm (exceptionally) sending this to the whole list,
> because several people have been involved.
>
> Look at this:
>
> %S A033188 1,1,2,6,6,30,150,210,210,210,2310,2310,30030,30030,30030,30030,
> %T A033188 510510,510510,9699690,9699690,9699690,9699690,223092870,223092870,
> %U A033188 223092870,223092870,223092870,223092870,6469693230,6469693230
> %N A033188 (Conjectural) minimum difference of an increasing arithmetic 
> progression of n primes.
> %D A033188 Computed by David W. Wilson
> ...
> %F A033188 David W. Wilson conjectures that a(n) = n# (n primorial, A034386) 
> for n >= 8.
> %A A033188 Eric W. Weisstein (eric(AT)weisstein.com)
>
> and
>
> %S A033189 
> 2,2,3,5,5,7,7,199,199,199,60858179,147692845283,14933623,834172298383,
> %T A033189 
> 894476585908771,1275290173428391,259268961766921,1027994118833642281
> %N A033189 Smallest first term of arithmetic progression of n primes with 
> difference A033188(n).
> %C A033189 a(18) found by Gusev and Andersen.
> ...
> %A A033189 Eric W. Weisstein (eric(AT)weisstein.com)
> %E A033189 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
> %E A033189 a(14) corrected by Gennady Gusev, Jul 07 2004
> %E A033189 Oct 05 2004: Gennady Gusev (gennady.gusev(AT)npo-saturn.ru) reports 
> that Jens Kruse Andersen found a(15), and Gennady Gusev and Jens Kruse 
> Andersen together found a(16) and a(17).
>
>
> In view of the great interest in these matters, I think
> it is important that these two sequences say which terms are
> known to be correct and which are just conjectures.
>
> Can David or Eric or Hugo or ... respond?
>
> Thanks
>
> Neil