Sequences needing more terms: high-value targets?

[Joseph S. Myers]

On Mon, 14 May 2007, Jon Schoenfield wrote:

> A related question:  the definitions of keywords "hard" and "more" at the OEIS
> web site both ask, "Would someone please extend this sequence?" ... but is
> there a conflict between them?  There are currently 1000 sequences with both:
> 
>    hard: Next term is not known and may be hard to find.
>    more: More terms are needed and should not be difficult to find.

Yes, I pointed out this in my message of Tue, 24 Feb 2004 and NJAS said:

    about "more" and "hard", - i agree with what
    Joseph says.  on the other hand it is all
    i can do keep up with the flow of new sequences and updates,
    so unless there are actual errors in the keywords i regard this
    matter as of low priority!

What I said in my original message was:

The keyword "hard" is defined as

     * hard: Next term is not known, and may be hard to find. Would
       someone please extend this sequence?

and "more" is defined as

     * more: More terms are needed and should not be difficult to find.
       Would someone please extend this sequence?

Lots of sequences have both keywords.  The definitions don't literally
contradict each other, but the combination suggests a lot of uncertainty
about how hard the extension is, and why say "hard" if you think it
shouldn't be hard but just "may" be?

I think there are actually three different cases where extensions are 
wanted:

* "hard" for genuinely hard sequences, where an extension is wanted and
hard to find and should be added even if it takes the sequence over the
normal length limit.

  [now, a b-file could be added for such an extension]

* "more" for short sequences that should be easy to extend (but don't need 
extending beyond the normal limit).

* Automatic detection of sequences significantly below the length limit
that should be extended and haven't had a considered judgement on how hard
they would be to extend.  Without a human judgement on the difficulty,
there's no need for a keyword, as the length present is easy for a
computer to measure.

(So the list of sequences that need extending would be split into the 
different types.)

-- 
Joseph S. Myers
jsm at polyomino.org.uk



The sum in question appears to be asymptotic to:




And the integral appears to converge (as N -> infinity) to:



(All of which assumes that I trust the PARI/GP implementation of numeric
integration...)