1,1

The aliquot sequence for n is the trajectory of n under repeated application of the map x -> sigma(x) - x.

The trajectory will either have a transient part followed by a cyclic part, or will have an infinite transient part and never cycle.

Sequence gives (length of transient part of trajectory) + (length of cycle (which is 1 if the trajectory reached 0)).

Examples of trajectories:

1, 0, 0, ...

2, 1, 0, 0, ...

3, 1, 0, 0, ... (and similarly for any prime)

4, 3, 1, 0, 0, ...

5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

6, 6, 6, ... (and similarly for any perfect number)

8, 7, 1, 0, 0, ...

9, 4, 3, 1, 0, 0, ...

12, 16, 15, 9, 4, 3, 1, 0, 0, ...

14, 10, 8, 7, 1, 0, 0, ...

25, 6, 6, 6, ...

28, 28, 28, ... (the next perfect number)

30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, ...

42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, ...

From this we may obtain many sequences:

Length of transient part + length of cycle: this sequence. Other versions of the current sequence: A044050, A003023

Length of transient part: A098008, also A007906. Records for transients: A098009, A098010.

Numbers which eventually reach 1 (or equivalently 0): A080907.

Aliquot trajectories for certain interesting starting values: A008885 A008886 A008887 A008888 A008889 A008890 A008891 A008892 A014360 A014361 A074907 A014362 A045477 A014363 A014364 A014365 A074906

For n<220 A098008 = A098007 - 1, i.e. 220 is the first sociable number. - Robert G. Wilson v, Sep 10 2004

Comment from T. D. Noe, Jun 06 2006: Up to 1000 there are 12 numbers whose fate is unknown, namely five well-known hard cases: 276, 552, 564, 660, 966 and seven others: 306, on same trajectory as 276; 396, on same trajectory as 276; 696, on same trajectory as 276; 780, on same trajectory as 564; 828, on same trajectory as 660; 888, on same trajectory as 552; 996, on same trajectory as 660.

J.-P. Delahaye, Les inattendus mathematiques, Chapter 19, "Nombres amiables et suites aliquotes", pp. 217-229, Belin-Pour la Science, Paris 2004.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

R. K. Guy, Unsolved Problems in Number Theory, B6.

R. K. Guy and J. L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Christophe CLAVIER, Aliquot Sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Aliquot sequence

P. Zimmermann, Aliquot Sequences

Author?, List of "prinitive" numbers not known to terminate (Oct 19 2004: list begins 276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, ...) [This is not the full list of numbers not known to terminate - see Comments above]

f:=proc(n) local t1, i, j, k; t1:=[n]; for i from 2 to 50 do j:= t1[i-1]; k:=sigma(j)-j; t1:=[op(t1), k]; od: t1; end; # produces trajectory for n

g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; Table[ Length[ f[n]] - 1, {n, 100}] (from Robert G. Wilson v Sep 10 2004)

Sequence in context: A103893 A106448 A159909 this_sequence A007554 A139069 A071866

Adjacent sequences: A098004 A098005 A098006 this_sequence A098008 A098009 A098010

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Sep 09 2004

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com) and John W. Layman (layman(AT)math.vt.edu), Sep 10 2004

Concerning one of the previously unsolved cases, Robert G. Wilson v reports that 840 reaches 0 after 749 iterations. - Sep 10 2004