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Old 2009-03-20, 16:24   #1
philmoore
 
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"Phil"
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Default Aliquot sequence convergence question

Quote:
Originally Posted by R.D. Silverman View Post
Note: I make the same suggestion to those chasing aliquot sequences in the factoring forum. Unless they have a well-established goal, there are better uses for the CPU time.
The aliquot sequence chasers might be doing it for the sheer fun of it, as they get to combine a number of different factoring techniques in pursuit of the extension of sequences. There are a number of unresolved conjectures in this area (see Richard Guy's book, for example) and Guy and Selfridge have conjectured that "most" sufficiently large even numbers generate aliquot sequences that do not terminate. Perhaps the data generated by these people can help formulate a reasonable conjecture of what "most" means.
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Old 2009-03-20, 17:16   #2
R.D. Silverman
 
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Quote:
Originally Posted by philmoore View Post
The aliquot sequence chasers might be doing it for the sheer fun of it, as they get to combine a number of different factoring techniques in pursuit of the extension of sequences. There are a number of unresolved conjectures in this area (see Richard Guy's book, for example) and Guy and Selfridge have conjectured that "most" sufficiently large even numbers generate aliquot sequences that do not terminate. Perhaps the data generated by these people can help formulate a reasonable conjecture of what "most" means.

It is clear, from a mathematical point of view what 'most' means:
a set of density 1. Unfortunately, no amount of computation will
ever resolve this conjecture. On the other hand, I have suggested projects
for which computation CAN resolve the conjecture.
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Old 2009-03-20, 18:23   #3
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My question was how fast this density approaches 1 as N increases, for which I am not aware of any conjectures supported by data.
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Old 2009-03-20, 19:04   #4
R.D. Silverman
 
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Quote:
Originally Posted by philmoore View Post
My question was how fast this density approaches 1 as N increases, for which I am not aware of any conjectures supported by data.

Ah. You are looking for a counting function.


#{s < n | aliquot(s) converges)

This would be very difficult to ascertain; It is likely to be something
that is at least as slow as loglog n. I don't know if the necessary
techniques are known to even approach this question theoretically.
It might yield to ergodic methods; ask Terry Tao.
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