Let us define a sequence S(N):
a1= N,
a
_{n}= sum of all factors of a
_{n1},
excluding a
_{n1} itself. Most sequences either converge or begin to repeat.
S(12) → 12,16,15,9,4,3,1,1,1...
S(6) → 6,6,6,6,...
S(220) → 220,284,220,284,220,284,...
Try to evaluate S(50), S(80), S(99).
How about S(138)?
WILL IT CONVERGE TO 1,1,1,...?
Please comment!
(In reply to
computer solution by Charlie)
Each of these is the Aliquot sequence of a number. Many of the interesting things you noticed are here:
https://en.wikipedia.org/wiki/Aliquot_sequence
Some highlights:
6 and 28 are perfect numbers. You table ends before the next which is 496.
https://en.wikipedia.org/wiki/Perfect_number
220 and 284 are the first pair of amicable numbers.
https://en.wikipedia.org/wiki/Amicable_numbers
276 is the first of a few numbers with unknown end behavior, but it is conjectured they do become periodic at some point.

Posted by Jer
on 20180510 14:46:46 