[math-fun] Iteration question
Let f(n) = sigma(n) - n = sum of proper divisors of n. If you iterate f on positive integer n, some trajectories end at 0 (where you cannot continue, since 0 is not in the domain of f), some trajectories loop (e.g. 6 -> 6... or 220 -> 284 -> 220...). For other numbers, like 138, the trajectory seems to diverge. Can we prove that there exist divergent trajectories?
To my knowledge, this question is still entirely shrouded in mystery. There may have been advances I know not of, however. On Thu, Sep 29, 2016 at 3:11 PM, David Wilson <davidwwilson@comcast.net> wrote:
Let f(n) = sigma(n) - n = sum of proper divisors of n.
If you iterate f on positive integer n, some trajectories end at 0 (where you cannot continue, since 0 is not in the domain of f), some trajectories loop (e.g. 6 -> 6... or 220 -> 284 -> 220...). For other numbers, like 138, the trajectory seems to diverge.
Can we prove that there exist divergent trajectories?
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Have a look at "aliquot sequence" and "sociable numbers". en.wikipedia.org/wiki/Aliquot_sequence mathworld.wolfram.com/AliquotSequence.html www.ams.org/journals/mcom/1975-29-129/ S0025-5718-1975-0384669-X/S0025-5718-1975-0384669-X.pdf The AMS article (by RKGuy) discusses "drivers" for aliquot sequences. The sequence beginning with 276 seems to diverge; it's been computed for several hundred terms. Nothing proven though. Loops are interesting: There seem to be a lot of 2-loops (amicable pairs). There are a few 4-loops known, a 5-loop, and a 29(?)-loop. See HAKMEM for the situation in 1972. There are conditional formulas known for amicable pairs, of the shape "if A and B and f(A,B) are prime, then g(A,B) and h(A,B) is an amicable pair". Similar formulas exist for 3-loops, but the conditions are so-far unfulfilled. Rich ---------------- Quoting Allan Wechsler <acwacw@gmail.com>:
To my knowledge, this question is still entirely shrouded in mystery. There may have been advances I know not of, however.
On Thu, Sep 29, 2016 at 3:11 PM, David Wilson <davidwwilson@comcast.net> wrote:
Let f(n) = sigma(n) - n = sum of proper divisors of n.
If you iterate f on positive integer n, some trajectories end at 0 (where you cannot continue, since 0 is not in the domain of f), some trajectories loop (e.g. 6 -> 6... or 220 -> 284 -> 220...). For other numbers, like 138, the trajectory seems to diverge.
Can we prove that there exist divergent trajectories?
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