old sequences
Simon Colton
simonco at dai.ed.ac.uk
Mon Jan 24 11:37:58 CET 2000
Dear Seqfans,
John Conway wrote:
> I fear that if there continues to be no control on the sequences inserted,
> the value of the system will only decline with time.
Brendan McKay wrote:
> A good partial solution to this problem is to have more
> intelligence in the data base.
I agree with John that perhaps too many dull sequences are
finding their way into the encyclopedia. Some of you may know
that I have written a program which invents integer sequences.
In an hour it can invent around 750 seqeuences which aren't in
the encyclopedia. However, I've restricted myself to only
submitting those which (a) have such an obvious definition that
they should have been in the EIS a long time ago (such as
refactorables - A033950) or (b) I can prove something interesting
about (such as A046952 which I have showed were the squares of
highly composite numbers).
However, I can also see why it is difficult to reject well
defined sequences (to paraphrase Neil - it increases the chances
of something like the moonshine conjectures being thrown up -
something which John knows a lot about!)
Brendan McKay has suggested a program that performs more
intelligent lookups. I agree entirely with this. Of course,
superseeker works very hard to identify a sequence. I would
suggest an even harder working program that runs on the user's machine,
not AT+T's. Hence a Java applet would be ideal. I'm presently adapting
my program, HR, to try and invent a definition for a given
sequence. HR is now written in Java, and will be available on
a web page some time in February.
Here is a nice example:
You may not be aware that, if you put any four numbers into the
EIS, a,b,c,d such that 0<a<b<c<d<10, then it will give you an
answer to the question: starting with a,b,c,d, what's the next
in the sequence. However, there are at present (I think) two
exceptions:
No sequence starts 4,5,6,9,... and
no sequence starts 4,5,7,9,...
So I asked my program to invent definitions for these. The second
sequence was very easy: HR spotted that if you take primes and add
2, you get a sequence starting 4,5,7,9. The superseeker also noted
that 5,7,9 is odd primes + 2 (sequence A048974). I think that the
sequence of primes + 2 should probably be in the encyclopedia (and
I suppose I should claim it for HR!)
The first sequence was much more difficult. However, HR has come
up with an answer:
Use the binary expansion of an integer, n, to write it as:
n = 2^{a_1} + 2^{a_2} + ...
eg. 11 = 2^{0} + 2^{1} + 2^{3}
Then let b(n) = {a_1,a_2,...}
eg. b(11) = {0,1,3} [is there a definition for this set?]
Then HR invented the following definition:
Let n be such that exactly two divisors of n are #not# in b(n).
The first four numbers, n, with this definition are 4,5,6,9
eg. b(4) = {2}, so divisors 1 and 4 are not in b(4).
b(5) = {0,2}, so divisors 1 and 5 are not in b(5).
b(6) = {1,2}, so divisors 3 and 6 are not in b(6)
etc.
The sequence goes: 4,5,6,9,13,14,15,17,22,27,29,35,...
It would be nice to have closure in the EIS over sequences starting
a,b,c,d, but I'm not sure this sequence should be in the
encyclopedia (I would hope there is a more interesting sequence
starting 4,5,6,9 - any solutions?) However, if you were desperate
for a sequence starting 4,5,6,9 this might do. And it just might
help you spot the moonshine! My point is that more intelligence
can be used to find matching sequences - even those which are not
directly related to ones in the encyclopedia. And I am volunteering
to write the software! This solution was produced in about 45 mins,
but I'm writing algorithms to improve the efficiency greatly.
I hope the program will not be an alternative to the EIS, but
work alongside it, finding matches for sequences that would ordinarily
be considered dull (such as the one above). Then, the encyclopedia
could contain only the most interesting sequences.
I'll keep seqfans informed about when HR will be available (not
long now). I'd greatly appreciate any suggestions about this work.
All the best,
Simon Colton.
---------------------------------
http://www.dai.ed.ac.uk/~simonco/
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