Numbers With Names

[Simon Colton]

Dear Sequence Fans,

I would like to draw your attention to a new program called NumbersWithNames
which is available to use here:

http://www.machine-creativity.com/programs/nwn/

The program is written in Java, which means you should be able to run it 
through
your browser with no downloading, configuring, etc. There is a manual and some
technical details about the program at the above site.

With Neil's permission and endorsement, it contains a subsection of about 1000
sequences from the Encyclopedia. The sequences are all number types which
have names in the literature, such as primes, odds, lucky numbers, triangular
numbers, etc.

The program is a sort of add-on to the Encyclopedia and has many links with 
it - there
is a button which will open the Encyclopedia for you in another window. It is
not meant in any way to rival the Encyclopedia, but it has a little extra 
functionality:

Firstly, if you have some numbers you are interested in, it will give some 
properties
of those numbers. For example,  if you type in the numbers of my birthday: 
23,3,73,
it tells you that these numbers are prime, weak, panconsummate, etc. There are
all sorts of ways to explore the sequences. For instance, if you type in 
the letter 'n'
and press return, it will list all number types with a name beginning with 
n. If you type
in 'nice', it will list all the number types which have this keyword in the 
Encyclopedia.

Secondly, given a sequence you are interested in (which can be one of the 
1000, or
a new one you give to the program), NumbersWithNames will make some conjectures
about it. It works quite hard to discard the dull conjectures and gives you 
a rating of
how plausible it thinks each conjecture is. For instance, you may know that 
I'm
interested in refactorable numbers (A033950, number of divisors is itself a 
divisor, aka tau
numbers). Previous re-incarnations of the program have told me many things 
about refactorables,
and just yesterday I ran the online program and it made the conjecture that 
e-perfect numbers
(A054979 - sum of the exponential divisors of n is 2n) are even 
refactorable numbers.
This is true of the entries in the Encyclopedia for e-perfect numbers and 
I'm still trying to
work out whether the conjecture is true or not. I hope that if you use the 
program, it will
make some interesting conjectures about the sequences you are interested in.

Whenever we use the OEOIS or superseeker, we are hoping that it will make a 
conjecture
for us: that the sequence it finds in the Encyclopedia has an equivalent 
definition to the
one we just typed in. The possibility of such conjectures arising from the 
Encyclopedia
is one of its great attractions, and Neil has recorded some very 
interesting results arising
this way. With NumbersWithNames, we've tried to enhance this process.

I would be very grateful if you would give the program a go. Please let me 
know if you do,
especially if you find any interesting results with it, or if you have any 
problems with it. In
particular, I'm trying to compile a list of interesting links between 
sequences in the
Encyclopedia (whether produced by the program or not).

All the best,

Simon Colton.

-----
http://www.dai.ed.ac.uk/~simonco
http://www.aisb.org.uk