[seqfan] Re: conjectured list of divisibility sequences
Charles Greathouse
charles.greathouse at case.edu
Mon Jan 10 22:25:29 CET 2011
a(0) must be divisible by the lcm of all the other terms, so a(0) = 0
is always acceptable but there are other possibilities.
I agree that it would be good to include normalized versions when
needed as well as marking translated divisibility sequences.
Personally, I think that we need a new method for keeping track of the
various properties which can be attached to sequences, something which
makes the information more readily searchable without taking up too
much screen space. I wouldn't want to have
Index entries for sequences related to recurrence relations
Index entries for sequences related to homogeneous recurrence relations
Index entries for sequences related to linear homogeneous recurrence relations
Index entries for sequences related to polynomials
Index entries for sequences related to periodic sequences
Index entries for sequences related to divisibility sequences
Index entries for sequences related to constant sequences
and I'd like to have a way to distinguish between "is related to" and
"is a member of". That would in turn allow for redundant information
not to be stored, or just not shown, depending on what is most
convenient.
Example 1: Information could be stored in keywords, with new keywords
like per for periodic sequences. Simple relationships could, perhaps,
be coded so that -keyword:per would also remove keyword:cons
sequences. Alternatively, every sequence with cons would also have
per, but this information could be hidden [by default?].
Example 2: Each sequence could have an associated bitfield with, say,
128 bits allowing the same information as #1 (but almost surely
recording each of the qualities separately -- space is cheap!).
Searching would be very fast and display could be handled as #1.
Example 3: Extended keywords, like base(10) and cons(conjectured).
Format remains the same as now, and searching could be either broad,
like "keyword:base", or narrow, like "keyword:base(10)".
Example 4: Categories, like in the aborted wiki version of the
sequences. Information is stored somehow, and sequences could be
browsed by category. These could be separate from or together with
keywords; I'd prefer together.
Example 5: Index links, like now, but with some distinction between
members and others. A sequence that is "related to polynomials" need
not be "related to linear recurrence relations", but a sequence that
*is* a polynomial, *is* a linear recurrence relation.
Just some thoughts about how we can make qualities like "divisibility
sequence" as convenient to work with (when editing as well as
browsing!) as the current keywords are.
Charles Greathouse
Analyst/Programmer
Case Western Reserve University
On Mon, Jan 10, 2011 at 3:08 PM, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:
> Dear All,
> I'm delighted in the current interest being shown
> about divisibility sequences. Is it not a fact that all
> such sequences have a(0) = 0 ? Also that all members of
> such a sequence are divisible by a(1), so that, whenever
> such a sequence has a(1) > 1, it is desirable also to
> list its normalized form with a(1) = 1. This way, people
> won't miss important coincidences. I've never understood
> offsets, but it also seems desirable to include (perhaps it
> already appears?) sequences such as 1,1,2,3,5,8,13,21, ...
> (# of ways of tiling a 2 by n box with dominoes), which
> is not a divisibility sequence until you translate it suitably.
>
> (Singular) sequences with zeroes should also appear.
> Aren't 0,0,1,1,0,0,1,1,0,0,... and lots of similar
> sequences, which often appear in the real world, already
> in there?
>
> A paper with a three-parameter family of fourth order
> divisibility sequences, infinitely many of which could
> legitimately appear in OEIS, by Hugh Williams and me,
> is due to appear in Internat J Number Theory. Best, R.
>
> On Mon, 10 Jan 2011, Richard Mathar wrote:
>
>>
>> http://list.seqfan.eu/pipermail/seqfan/2011-January/006837.html
>>
>> pl> Date: Mon, 10 Jan 2011 12:05:52 +0100
>> pl> From: "peter.luschny"
>> pl> To: seqfan at list.seqfan.eu
>> pl> Subject: [seqfan] Re: conjectured list of divisibility sequences
>> pl>
>> pl> I see that A000045 is not in your list and it is also
>> pl> not in the OEIS index of divisibility sequences. Do I
>> pl> misunderstand your definition?
>>
>> The Fibonacci sequence A000045 (and any sequnce a= F(n)^k, k>=1) is
>> a divisibility sequence which is not in the current list because I
>> excluded
>> sequences that contain zeros (as indicated in the small print of the web
>> page);
>> A000045(0)=0 caused it to be ignored. I'll improve this part because
>> not listening that "classic" example puts shame on the entire list.
>>
>> Richard Mathar
>>
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