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?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun