20090320, 16:24  #1 
"Phil"
Sep 2002
Tracktown, U.S.A.
3×373 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  
Nov 2003
2^{2}×5×373 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.
1119_{10} 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  
Nov 2003
1D24_{16} 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. 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Aliquot sequence reservations  schickel  Aliquot Sequences  3446  20210422 00:00 
Another Aliquot Sequence site  schickel  Aliquot Sequences  67  20120120 17:53 
Aliquot sequence worker for factordb  yoyo  FactorDB  6  20120112 20:58 
YA aliquotsequencechasing script  fivemack  Aliquot Sequences  5  20090928 16:40 
Useful aliquotsequence links  10metreh  Aliquot Sequences  2  20090731 17:43 