A003023 "Length" of aliquot sequence for n.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

See A098007 for a better version.

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

Examples

			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, ...
		

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. 557-580 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

• Antti Karttunen, Table of n, a(n) for n = 1..275 (fate of 276 is unknown)
• 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. [Annotated scanned copy]
• F. Richman, Aliquot series: Abundant, deficient, perfect
• Eric Weisstein's World of Mathematics, Aliquot Sequence
• Wikipedia, Aliquot sequence

Crossrefs

Cf. A001065, A098007.

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

Programs

Maple

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

Mathematica

f[x_] := (k++; DivisorSigma[1, x] - x); f[1] = 1;
Table[k = 0; FixedPoint[f, n]; k, {n, 1, 102}]
(* Jean-François Alcover, Jul 27 2011 *)

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; i-1 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)))))))
(define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry.
;; Antti Karttunen, Nov 03 2017

Extensions

More terms from Matthew Conroy, Jan 16 2006