A063769

[jb at brennen.net]

> 1) there are a lot of numbers whose aliquot sequences get pretty big.
> How big should they have to get before I go along with the comment in
> the existing sequence about 276, that it "seems unlikely" that they'll
> ever reach a perfect number?  I arbitrarily took 10^7 as my bound, but
> that left about 10% of numbers (1074 numbers under 10,000) hitting the
> bound.  So I tried 10^8 and found 1065 --- increasing that bound by a
> factor of 10 didn't help much.  I suppose I could use a more efficient
> factorization algorithm [right now I just check every number up to the
> square root] and try bigger bounds, but it sure looks like a lot of
> these are headed to the stratosphere...

Yeah, you need a more efficient means of factoring.  :)

Without trying too hard, I found that 84444 reaches as high as
99543472803136 before coming down to the perfect number 8128.

Or the number 90360, which goes up to 1071102085964 before
finally ending up at 6.

It would be a reasonable conjecture that the proportion of numbers
whose aliquot sequence "goes perfect" might actually approach a
constant.  There are some good reasons to believe that this constant
exists and that it is not vanishingly small.  If this is true, it
means that there exist many large numbers which end up at a
perfect number.  Yeah, it's unlikely that 276 ends up at a perfect
number -- over any reasonable sample size, less than 1% of starting
numbers do so.  But generalizing this to say that all numbers whose
aliquot sequence gets "large" do not end up at a perfect number is
not justified.

Also, if you do any sort of systematic categorization of aliquot
sequences, make sure that you take steps to avoid doing the same
work over and over.  A few years ago, I completed the "unitary"
aliquot sequences for all of the integers up to 4*10^8, and showed
that all of them terminated.  A big help is remembering that once
you have computed a sequence to termination, you don't have to
do any work on any of the members of that sequence ever again.

For instance, if you complete the sequence for 84444 and find that
it terminates in a perfect number, go back and mark that 119844
(the successor to 84444) also terminates in a perfect number.
Etc., etc., etc.


What I've done in the past is something like this:  Compute each sequence
until it either terminates in a loop or exceeds 10^10.  In either
case, mark all members of that sequence as being done for now.  Once
you're done, you'll have a big list of numbers whose sequences have
been completely computed, a smaller list of numbers whose sequences
have been computed up to a point in excess of 10^10, and an even
smaller list of numbers > 10^10 which need to be "finished".  Then
you take that list of numbers > 10^10 and compute each one until it
either terminates or exceeds 10^15.  And so on, and so on, you keep
raising the threshold until there are only a few sequences left
uncomputed.


   Jack