

A003023


"Length" of aliquot sequence for n.
(Formerly M0062)


10



0, 1, 1, 2, 1, 1, 1, 2, 3, 3, 1, 6, 1, 4, 4, 5, 1, 3, 1, 6, 2, 5, 1, 4, 2, 6, 2, 1, 1, 14, 1, 2, 5, 7, 2, 3, 1, 6, 2, 3, 1, 13, 1, 4, 6, 7, 1, 5, 3, 2, 3, 8, 1, 12, 2, 4, 2, 3, 1, 10, 1, 8, 2, 3, 2, 11, 1, 4, 3, 5, 1, 8, 1, 4, 4, 4, 2, 10, 1, 6, 4, 5, 1, 5, 2, 8, 6, 6, 1, 9, 3, 5, 3, 3, 3, 8, 1, 2, 3, 4, 1, 17
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OFFSET

1,4


COMMENTS

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)  1 (if trajectory ends at 1), or provided that it ends in cycle [e.g., (6) or (220 284)], gives (length of transient part of trajectory) + (length of cycle) = length of trajectory.  Corrected by Antti Karttunen, Nov 03 2017
The function sigma = A000203 is defined only on the positive integers and not for 0, so the trajectory ends when 0 is reached.  M. F. Hasler, Nov 16 2013


REFERENCES

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. 557580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

R. K. Guy and J. L. Selfridge, Interim report on aliquot series, pp. 557580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971. [Annotated scanned copy]


EXAMPLE

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, ...


MAPLE

f:=proc(n) local t1, i, j, k; t1:=[n]; for i from 2 to 50 do j:= t1[i1]; k:=sigma(j)j; t1:=[op(t1), k]; od: t1; end; # produces trajectory for n


MATHEMATICA

f[x_] := (k++; DivisorSigma[1, x]  x); f[1] = 1;
Table[k = 0; FixedPoint[f, n]; k, {n, 1, 102}]


PROG

(MuPAD) s := func(_plus(op(numlib::divisors(n)))n, n): A003023 := proc(n) local i, T, m; begin m := n; i := 1; while T[ m ]<>1 and m<>1 do T[ m ] := 1; m := s(m); i := i+1 end_while; i1 end_proc:
(Scheme)
(define (A003023 n) (let loop ((visited (list n)) (i 0)) (let ((next (A001065 (car visited)))) (cond ((zero? next) i) ((member next visited) (+ 1 i)) (else (loop (cons next visited) (+ 1 i)))))))


CROSSREFS

Cf. A059447 (least k such that n is the length of the aliquot sequence for k ending at 1).


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



