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

[Hugo Pfoertner]

"N. J. A. Sloane" wrote:
> 
> 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

In the meantime Jens Kruse Andersen has answered the questions in an
e-mail he sent to Neil:

Hugo Pfoertner wrote:

> In the non-public Sequence Fanatic discussion forum (SeqFan) Neil
> Sloane asked for some clarification on the status of two OEIS
> sequences. Is there any feedback on this topic from the primeform
> community? Neil's message is attached.

I don't subscribe to SeqFan but somebody notified me of Neil's request.
I mailed the following to Neil.

The status of minimal AP's is maintained at my
http://hjem.get2net.dk/jka/math/aprecords.htm#minimal
Blue entries there are proven minimal (meaning somebody claims to have
made an
exhaustive search).
Red entries are the best currently known when the minimal is unknown.

The current status for A033188:
The value of a(1) is a matter of definition: What is the "difference" in
the
"arithmetic progression of length 1" consisting of the prime 2. I would
tend
to call it either 0 or undefined (my site avoids the question by not
listing
AP1).
a(2) to a(18) are proven.
a(n) for n>18 are conjectured values based on the plausible conjecture
a(n)=n#
(n primorial) for n>=8, which is currently proven for n<=18 and no
larger
value.

The current status for A033189:
a(1) to a(18) are proven (this is all the currently listed values).
There are
no conjectured values after a(18), where there are no known arithmetic
progressions with the conjectured difference in A033188.

The current status for A113872 is the same as for A033189:
a(1) to a(18) proven, and no conjectured values.
By the way, the comment has two links to A033189. One should be to
A033188.

A suggestion: Change A033188 to only list the known values, currently
ending
at a(18). Add a "hard" keyword and continue to mention the conjecture.

--
Jens Kruse Andersen