In an aliquot chain one starts with a number, calculates the sum of its proper divisors, and repeats the process on the result.
arithmetic/aliquot.sequences New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
In an aliquot chain one starts with a number, calculates the sum of its
proper divisors, and repeats the process on the result. Example:
20 (proper divisors 1, 2, 4, 5, 10)
22 (proper divisors 1, 2, 11)
14 (proper divisors 1, 2, 7)
10 (proper divisors 1, 2, 5)
8 (proper divisors 1, 2, 4)
7 (proper divisors 1)
1 (no proper divisors)
1) Do all chains end up either repeating or hitting 1, or do infinite,
non-repeating chains exist? The smallest chain "starter" where it
is unknown how the chain ends is 276 (or has this been solved?).
2) Are there infinitely many different repeating chains?
3) Which cycle lengths of repeating chains are possible? (I know the
"usual" perfect numbers, 500+ chains of length 2, 14 chains of
length 4, 1 of length 5, and 1 of length 28.)
-- Jan Otto Munch Pedersen <jomp@mentor.dk>, June 1992
Show solution
[ Editor: on question 1: ]
According to Richard Guy, there is strong heuristic evidence that
almost all chains starting from even numbers increase to infinity, but
this is unproven. See:
R K Guy, Aliquot sequences, in Number Theory and Algebra,
Academic Press, 1977, 111--118.
R K Guy and J L Selfridge, What drives an aliquot sequence?
Math. Comput. 29 (1975), 101--107.
As of the fall of 1990, the fate of 276 is still unknown.
[ Editor: on questions 2 and 3: ]
It has been conjectured that length 3 does not occur. It has
also been conjectured that infinitely many cycles occur of any desired
length. This is surely true for lengths 1 and 2 and I imagine it to be
true for length 4 as well. Nothing has been proven.
Tens of thousands of length 2 cycles (`amicable pairs') are known. I
know of 39 cycles of greater length, 33 of length 4, 2 of length 8,
and one each of lengths 5, 6, 9, and 28. The following are the
smallest elements of these cycles.
#1, 12496 (length 5); #2, 14316 (length 28); #3, 1264460; #4, 2115324;
#5, 2784580; #6, 4938136; #7, 7169104; #8, 18048976; #9, 18656380;
#10, 28158165; #11, 46722700; #12, 81128632; #13, 174277820; #14, 209524210;
#15, 330003580; #16, 498215416 (all length 4); #17, 805984760 (length 9);
#18, 1095447416 (length 8); #19, 1236402232 (length 4);
#20, 1276254780 (length 8); #21, 1799281330; #22, 2387776550; #23, 2717495235;
#24, 2879697304; #25, 3705771825; #26, 4424606020; #27, 4823923384;
#28, 5373457070; #29, 8653956136; #30, 15837081520;
#31, 17616303220 (all length 4); #32, 21548919483 (length 6);
#33, 21669628904; #34, 88585861815; #35, 1092162882824; #36, 1755676229313195;
#37, 114588454336625295; #38, 127735111770308496156105;
#39, 455449879323655623656384 (all length 4).
#1, #2 from
P Poulet, L'intermediare des math. 25 (1918), 100--101.
#3--#9, #11--#16 from
H Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), 423--429.
J S Devitt, R K Guy, and J L Selfridge, Third report on aliquot sequences,
Congr. Numer. XVIII, Proc. 6th Manitoba Conf. Numerical Math. 1976,
pp. 177--204.
S C Root, Item 61, MIT Artificial Intelligence Memo no. 239, Feb. 29, 1972.
#10 from
W Borho, Ueber die Fixpunkte der k-fach iterierten Teilersummenfunktion,
Mitt. Math. Gesellsch. Hamburg 9 (1969), 34--48.
#17--#27 from
A Flammenkamp, New sociable numbers, Math. Comp. 56 (1991), 871--873.
#28, #29, #34--#39 from
D Moews and P C Moews, A search for aliquot cycles below 10^{10},
Math. Comp. 57 (1991), 849--855.
#30--#33 from
D Moews and P C Moews, A search for aliquot cycles and amicable pairs,
preprint.
Also, Ren Yuanhua independently found #22, #34, and #37.
-- David Moews <moews@math.berkeley.edu>, 1992
[ Editor: the cycles of length 1 are the perfect numbers; see
arithmetic/perfect.numbers. ]