20090320, 16:24  #1 
"Phil"
Sep 2002
Tracktown, U.S.A.
2140_{8} Posts 
Aliquot sequence convergence question
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.

20090320, 17:16  #2  
"Bob Silverman"
Nov 2003
North of Boston
2·3^{3}·139 Posts 
Quote:
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. 

20090320, 18:23  #3 
"Phil"
Sep 2002
Tracktown, U.S.A.
2^{5}×5×7 Posts 
My question was how fast this density approaches 1 as N increases, for which I am not aware of any conjectures supported by data.

20090320, 19:04  #4  
"Bob Silverman"
Nov 2003
North of Boston
2·3^{3}·139 Posts 
Quote:
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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