subsequences

[Simon Colton]

Sorry to bother you all again,

Brendan McKay wrote:

> [Aside: a search option that looks for sequences containing
> a set of numbers but not in any particular order would be
> easy to add and quite useful I think.]

Again, I agree. I've used such a datamining technique to
find sequences in the EIS which are subsequences of those
invented by HR. The most appealing result so far is that
A023194 is a subsequence of A009087 (invented by HR). This
translates to the theorem:

Given an integer, n, if the sum of divisors of n is prime
then the number of divisors of n will be prime.

ie. forall n, tau(sigma(n))=2 -> tau(tau(n))=2.

Has anyone seen this result before? Not amazingly difficult
to prove, but a nice result nevertheless.

Such a subsequence search is computationally expensive, and
needs lots of control. Again, it might be better to have this
run on the user's machine. There are some other datamining 
techniques which can be employed, such as finding two disjoint
sequences. This helped me notice that perfect numbers are not
refactorable (see JIS paper). I've just submitted another paper 
about this datamining work to a conference on Artificial Intelligence
(my area), if anyone is interested.

Cheers,

Simon
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http://www.dai.ed.ac.uk/~simonco/